422 2 Lecture Notes for EC422 Mathematical Economics 2 1 _Integral

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					               .422                      2




Lecture Notes for EC422 Mathematical Economics 2




                              1
                            (Integral Calculus)




                        .




                            2553
                                                                                                        2




                                                             (Differential Calculus)
                     (Differentiation)                        (Derivatives)

                                                                        (Integral Calculus)
                                    (Antidifferentiation)                                      (Indefinite
Integration)                           (Antiderivative)                                (Indefinite Integral)




                                   ⇐⇒
                                   ⇐⇒
                                   ⇐⇒


1.1                                        (Indefinite Integral)
                                                                          dF
         1.1.1.               f                                F             (x) = f (x)
                                                                          dx
F                                 (antiderivative)                 f                           (indefinite
integral)                 f

            1.1.2.                                   f (x)



    f (x) = 2x                              F (x) = x2 ,                      F (x) = x2 + 4
    f (x) = 3                               F (x) = 3x,                       F (x) = 3x + 8
    f (x) = 0                               F (x) = 5,                        F (x) = −1
    f (x) = ex                              F (x) = ex ,                      F (x) = ex + 1
            1
    f (x) = , x > 0                         F (x) = ln x,                     F (x) = ln x − 4, x > 0
            x
              1                                     √                                 √
    f (x) = √ , x > 0                       F (x) = x,                        F (x) = x − 2, x > 0
            2 x
                                                      1                                 1
    f (x) = x−2 , x = 0                     F (x) = − ,                       F (x) = − + 1, x = 0
                                                      x                                 x
    f (x) = cos x                           F (x) = sin x,                    F (x) = sin x − 5



                                                                                    dF
            :                                                                  F       (x) = f (x)
                                                                                    dx
                                                                                                              3


               :                       . . . dx                                                (indefinite in

tegration operator)                                                                                f
                   F (x) =       f (x)dx

               1.1.3.        f           g                  f (x) = g(x)                    x∈R

       c                F (x) = G(x) + c

                             H(x) = F (x) − G(x)                                              h(x) = f (x) −
g(x) = 0                         H(x) = c                          c                  c = F (x) − G(x)
F (x) = G(x) + c


                                                                                                           . . .


               1.1.4.                    F                                    f
        f                                     F (x) + c        c

                                                                                  dG
                             G                                         f             (x) = g(x) = f (x)
                                                                                  dx
                                                  dF
F                                  f                 (x) = f (x) = g(x)                            1.1.3
                                                  dx
       c                G(x) = F (x) + c

                                                                                                           . . .

           :            c                                              (arbitrary constant of integration)



                                                                                             1.1 – 1.3
                                                                                                   4


                              (Differentiation)                     (Integration)
                   da
                        =     0                           0 dx      = c
                   dx
                   dx
                        =     1                           1 dx      = x+c
                   dx
                 dxn                                                        xn+1
                        = nxn−1                          xn dx      =            +c
                  dx                                                        n+1
                  dex
                        = ex                              ex dx     = ex + c
                  dx
                 dax                                                         ax
                        = ax ln a, a > 0                 ax dx      =            +c
                  dx                                                        ln a
               d ln x         1                           1
                        =                                   dx      =       ln |x| + c
                dx            x                           x
            d loga x             1                      1
                        =            ,a > 0                 dx      =       loga x + c
               dx             x ln a                 x ln a

                 1.1:



                    (Differentiation)                                       (Integration)
       d sin x
                  =     cos x                            cos x dx =           sin x + c
         dx
       d cos x
                  =     − sin x                          sin x dx = − cos x + c
         dx
      d tan x
                  =     sec2 x                          sec2 x dx       =     tan x + c
         dx
       d sec x
                  =     sec x tan x               sec x tan x dx =            sec x + c
         dx
       d csc x
                  =     − csc x cot x              csc x cot x dx       = − csc x + c
         dx
       d cot x
                  =     − csc2 x                        csc2 x dx       = − cot x + c
         dx
 d arcsin x                     1
                            √
      dx          =           1 − x2                      1                       arcsin x + c
 d arccos x                        1                √          dx       =
                  =         −√                          1 − x2                    − arccos x + c
     dx                          1 − x2
 d arctan x                    1
     dx           =         1 + x2                        1                       arctan x + c
d arccot x                        1                            dx       =
                  =         −                           1 + x2                    −arccot x + c
     dx                       1 + x2
d arcsec x                        1
                               √
     dx           =         |x| x2 − 1                 1                          arcsec x + c
d arccsc x                           1               √       dx         =
                  =         − √                   |x| x2 − 1                      −arccsc x + c
     dx                       |x| x2 − 1

     1.2:
                                                                                                 5


                    (Differentiation)                                        (Integration)
     d sinh x
                =      cosh x                               cosh x dx    =   sinh x + c
        dx
    d cosh x
                =      sinh x                               sinh x dx    =   cosh x + c
        dx
   d tanh x
                =      sech2 x                              sech2 x dx   =   tanh x + c
       dx
   d sech x
                =      −sech x tanh x             sech x tanh x dx       = −sech x + c
       dx
   d csch x
                =      −csch x coth x             csch x coth x dx       = −csch x + c
       dx
    d coth x
                =      −csch2 x                             csch2 x dx   = − coth x + c
        dx
  d sinh−1 x               1                                1
                =      √                                √        dx      =   sinh−1 x + c
      dx                 1 + x2                           1 + x2
 d cosh−1 x                1                                1
                =      √                                √        dx      =   cosh−1 x + c
      dx                 x2 − 1                          x2 − 1
d tanh−1 x                 1
                =              , |x| < 1                      1                 tanh−1 x + c, |x| < 1
     dx                 1 − x2                                     dx    =
d coth−1 x      =          1
                               , |x| > 1                    1 − x2              coth−1 x + c, |x| > 1
     dx                 1 − x2
 d sech−1 x                   1                           1
                =      − √                              √       dx       = −sech−1 |x| + c
      dx                 |x| 1 − x2                  |x| 1 − x2
 d csch−1 x                   1                           1
                =      − √                              √       dx       = −csch−1 |x| + c
      dx                 |x| 1 + x2                  |x| 1 + x2

       1.3:




         1.1.5. (                        )
     (linear operator)                        f         g                                    a




1.    af (x) dx = a      f (x) dx


2.    f (x) + g(x) dx =          f (x) dx +   g(x) dx
                                                                                                                   6




          d                                    d
               a           f (x) dx       = a                   f (x) dx      = af (x)
         dx                                  dx
 d                                           d                                  d
      f (x) dx +           g(x) dx        =                    f (x) dx +                 g(x) dx     = f (x) + g(x)
dx                                          dx                                 dx

                                                                                                               . . .

      1.1.6.



                                                                                               x2
 1.   (3x + 5)dx =              (3x)dx +     5 dx = 3                 x dx + 5      1 dx = 3      + c1 + 5x + c2
                                                                                               2
                        3
                       = x2 + 5x + c
                        2
                   2
      √
      3          x 3 +1      3 5
 2.     x 2 dx =        + c = x3 + c
                 2
                 3
                   +1        5
                   2
           1                                      1                                                   1
 3.     x+             dx =            x2 + 2 +                dx =       x2 dx + 2       1 dx +         dx
           x                                      x2                                                  x2
                                 x3       1
                            =       + 2x − + c
                                 3        x
        1     1                    1                   1       1  4 3
 4.       4
            +√              =         dx +            √ dx = − 3 + x 4 + c
        x    4
               x                   x4                 4
                                                        x     3x  3

 5.   (2x3 − 5x2 − 3x + 4)dx = 2                      x3 dx − 5           x2 dx − 3       x dx + 4     1 dx
                                          1    5    3
                                         = x4 − x3 − x2 + 4x + c
                                          2    3    2
         2     2       3                          4    2          2   4
 6.    a3 − x3             dx =        a2 − 3a 3 x 3 + 3a 3 x 3 − x2 dx
                                                           4      2           2       4
                                = a2     1 dx − 3a 3             x 3 dx + 3a 3      x 3 dx −       x2 dx
                                        9 4 5 9 2 7 1
                                = a2 x − a 3 x 3 + a 3 x 3 − x 3 + c
                                        5         7         3
                            3                                                 3                       1 3
 7.     5ex − x−2 +               dx = 5     ex dx −              x−2 dx +          1 dx = 5ex +       + x+c
                            2                                                 2                       x 2
                                                                                                                           7


1.2                                       (Definite Integral)


           (bounded function) f (x)                           x                       x = a         x = b
               x1 , x2 , . . . , xn−1                                        [a, b]             n                a = x0 <

x1 < x2 < · · · < xn−1 < xn = b
  0                                             (n → ∞)

∆xi = xi − xi−1              i = 1, 2, . . . , n              lim           max ∆xi      =0
                                                            n→∞         1≤i≤n

                       i = 1, 2, . . . , n                        fi                          (infimum)           f (x)
      xi−1 ≤ x ≤ xi                     fi =      inf       f (x)
                                               xi−1 ≤x≤xi
                                                                                                           n
                                                (Lower Riemann Sum)                                 Sn =         f i ∆xi
                                                                                                           i=1

               1.1




                                           1.1:



                           i = 1, 2, . . . , n                         fi                      (supremum)              f (x)
        xi−1 ≤ x ≤ xi                     fi =          sup       f (x)
                                                   xi−1 ≤x≤xi
                                                                                                                  8

                                                                                                       n
                                                    (Upper Riemann Sum)                       Sn =           f i ∆xi
                                                                                                       i=1

                    1.2




                                             1.2:



           A                         f (x)               [a, b]               Sn ≤ A ≤ Sn
                                    [a, b]                             n

                                                                                                    S1 ≤ S2 ≤
. . . S n−1 ≤ S n ≤ S n+1 ≤ . . .             (S n )∞
                                                    n=1                              (nondecreasing sequence)
                  (upper bound)                     A                   (S n )∞
                                                                              n=1
                                                                              b
                                       (Lower Riemann Integral)                   f (x) dx = lim S n
                                                                                              n→∞
                                                                              a
                                                           b
A                                                              f (x) dx ≤ A
                                                           a
                                                         [a, b]                           n


        S 1 ≥ S 2 ≥ . . . S n−1 ≥ S n ≥ S n+1 ≥ . . .                         (S n )∞
                                                                                    n=1

(nonincreasing sequence)                                (lower bound)               A                  f (x)
                    [a, b]              S1 < ∞                          (S n )∞
                                                                              n=1
                                                                                                                                      9

                                                                                           b
                                           (Upper Riemann Integral)                                f (x) dx = lim S n
                                                                                           a                     n→∞
                                                               b
A                                                                  f (x) dx ≥ A
                                                               a
                                                                   b                                   b
                                                                       f (x) dx ≤ A ≤                      f (x) dx
                                                                   a                                   a
    b                  b                                                           b                                   b
        f (x) dx =         f (x) dx                                                    f (x) dx = A =                      f (x) dx
    a                  a                                                           a                                   a
                                                                                           b
                                             (Riemann Integral) A =                            f (x) dx
                                                                                       a
                                                                          (Riemann Integration)
                                                                                           f



(Riemann Integrable)

            1.2.1. (                                                            )                  Q
(Rational Number)


                                                       1                  x∈Q
                                           f (x) =
                                                       0                  x∈Q
                                                                           /



                                  [0, 1]                                            n
             0                                             1                fi = 0                          fi = 1

i = 1, 2, . . . , n                                                        Sn = 0
Sn = 1                                                                                         (                             n)
                                      1
                                          f (x) dx = lim S n = 0
                                                      n→∞
                                      0
              1
                  f (x) dx = lim S n = 1                                       f
              0                n→∞




                                                       (Antiderivative)
                                                     (Riemann Integral)
                                                                                                                                 10




                                                                                                                  (Fundamental

Theorem of Calculus)

         1.2.2.                                               (Fundamental Theorem of Calculus)
   f                                                                 [a, b]           F                                      f



                                               b
                                                   f (x) dx = F (b) − F (a)
                                           a

                                  [a, b]                  n                  a = x0 < x1 < x2 < · · · < xn−1 <
xn = b                       i = 1, 2, . . . , n                                 [xi−1 , xi ]             ∆xi = xi − xi−1

         F                                 f                         F                                          (differentiable
everywhere)                          F             f                                            (Mean Value Theorem)
                              i = 1, 2, . . . , n                    xi ∈ [xi−1 , xi ]
                                                                     ˜



                        F (xi ) − F (xi−1 ) = F (˜i )(xi − xi−1 ) = f (˜i )∆xi
                                                 x                     x

              xi ∈ [xi−1 , xi ]
              ˜                                f i ≤ f (˜i ) ≤ f i
                                                        x



                                   f i ∆xi ≤ F (xi ) − F (xi−1 ) ≤ f i ∆xi

                                         i = 1            n                                  n → ∞


                        n                          n                                   n
                             f i ∆xi ≤                 F (xi ) − F (xi−1 )        ≤         f i ∆xi
                       i=1                      i=1                                   i=1
                                  Sn ≤                 F (b) − F (a)              ≤ Sn
         b                                                                                                b
             f (x) dx = lim S n ≤                      F (b) − F (a)              ≤ lim S n =                 f (x) dx
                            n→∞                                                       n→∞                 a
         a

                                                                         b                      b                      b
         f                                                                   f (x) dx =             f (x) dx =             f (x) dx
                                                                         a                      a                  a
                                                                                                               11

                               b
                                   f (x) dx = F (b) − F (a)
                           a

                                                                                                           . . .



 (Definite Integration)

         1.2.3.



         5                 5
1.           3x2 dx = x3       = 125 − 1 = 124
     1                     1

         b                     b
2.           kex dx = kex          = k(eb − ea )
     a                         a

         4
               1                                     4
3.               + 2x dx = ln |x| + x2                   = ln 4 + 16 − ln 1 − 1 = ln 4 + 15
     1         x                                     1

         2                                    2                                     2
4.           (2x3 − 1)2 (6x2 )dx =                (4x6 − 4x3 + 1)(6x2 )dx =             24x8 − 24x5 + 6x2 dx
     1                                    1                                     1
                                                                     2                         2
                                        24 9 24 6 6 3           8
                                      =    x − x + x = x9 − 4x6 + 2x3
                                         9       6     3 1      3                              1
                                        8 × 512                   8
                                      =         − (4 × 64) + 16 − + 4 − 2
                                            3                     3
                                        4088               2            2
                                      =       − 238 = 1362 − 238 = 1124
                                          3                3            3
                                                                                                                                                 12


                    1.2.4.
                b                                a
   1.               f (x) dx = −                     f (x) dx
            a                                b
                a
   2.               f (x) dx = 0
            a

                d                        b                                    c                          d
   3.               f (x) dx =               f (x) dx +                           f (x) dx +                 f (x) dx        a≤b≤c≤d
            a                        a                                    b                          c

                b                                        b
   4.               −f (x) dx = −                            f (x) dx
            a                                        a

                b                                    b
   5.               kf (x) dx = k                        f (x) dx
            a                                    a

                b                                                 b                        b
   6.                f (x) + g(x) dx =                                f (x) dx +               g(x) dx
            a                                                 a                        a


                                                                                                 (                              )

                                                                                                                                              . . .

                    1.2.4    5                   6                                                                                         (Definite
                                 b
Integral Operator)                   . . . dx                                                            (linear operator)
                             a



1.3                                                                     (Improper Integral)

                                                                        [a, b]                                          (bounded function)

                                                                                                                                              (upper
limit of integration)                                             +∞                                                                (lower limit of
integration)                     −∞

                                                     (unbounded function)                                                           (singular point)
                                         +∞                             −∞
                                                 (Improper Integration)                                                                         (

        )
                                                                                                                                                 13




                          ∞                                  b                             b                                      b
                              f (x) dx = lim                     f (x) dx                       f (x) dx = lim                        f (x) dx
                      a                            b→∞   a                                 −∞                      a→−∞   a
                      ∞                                      b                                                 b
                          f (x) dx = lim                         f (x) dx        =              lim                f (x) dx
                     −∞                           a→−∞   a                               (a,b)→(−∞,∞)      a
                                                   b→∞


                 :                                                                             (a, b)                                   (−∞, ∞)
                                                                                                                              b
                                                                                                         lim lim                  f (x) dx
                                                                                                         b→∞ a→−∞         a
                                                             b
                                      lim lim                    f (x) dx (                                                             )
                                  a→−∞ b→∞               a
                                 f                                          c ∈ [a, b]



         b                            c−                             b                          d                             b
             f (x) dx =                       f (x) dx +                 f (x) dx = lim−
                                                                                                    f (x) dx + lim+
                                                                                                                                  f (x) dx
     a                            a                               c+                 d→c    a                      d→c    d




             1.3.1.



             ∞                             ∞
                     1         1
1.                      dx = −                     =0+1=1
         1           x2        x           1
             ∞                             ∞
                     1
2.                     dx = ln |x|                 = ∞ − ln 1 = ∞ − 0 = ∞
         1           x                     1

             1
                 1                        1
3.                 dx = ln |x|                    = ln 1 − lim ln |x| = 0 − (−∞) = ∞
         0       x                        0+                  +  x→0

             9                                9
                      1               1                          1
4.               x− 2 dx = 2x 2                    = (2 × 9 2 ) − (2 × 0) = 2 × 3 = 6
         0                                    0+




                      (        ∞ − ∞)
                                                                                                              14


            1.3.2.



            ∞                          b                              b
  1.            sin x dx = lim             sin x dx = lim − cos x         = lim (1 − cos b)
        0                  b→∞     0                  b→∞             0     b→∞

            1              0−                 1                    0−             1
                1               1                 1         1          1                           1        1
  2.               dx =            dx +               dx = − x−2      − x−2            =    −∞ +       −      −∞
        −1      x3        −1    x3           0+   x 3       2      −1  2          0+               2        2
                     =∞−∞




                                           (Cauchy Principal Value Integral)




1.4                                                 (Integration Technique)

                                                                                                  (product rule)
                 (quotient rule)                       (chain rule)




            3                                                 (integration by substitution)

                     (integration by parts)                                            (partial fractions method)


1.4.1                                                    (integration by substitution)


                  f (x)                      φ(x)                       u = φ(x)                   x = φ−1 (u)
                                                        dφ−1
                                (differential) dx =          (u)du
                                                         du
                          x=a              u = φ(a)           x=b             u = φ(b)
                                                                                                                             15


                                   b                        φ(b)
                                                                                    dφ−1
                                       f (x) dx =                     f φ−1 (u)          (u)du
                               a                           φ(a)                      du



x = g(u)

           1.4.1.                                                         (integration by substitution)                  f
                             [a, b]               x = g(u)                   g

                 g(α) = a               g(β) = b

                                            b                         β
                                                                                   dg(u)
                                                f (x) dx =                f g(u)         du
                                        a                         α                 du
                         F                                        f


                  d(F ◦ g)(u)   dF g(u)   dF g(u) dg(u)          dg(u)
                              =         =               = f g(u)
                      du           du        dx    du             du




           β                                        g(b)
                        dg(u)                              d(F ◦ g)(u)
               f g(u)         du =                                     du = (F ◦ g)(β) − (F ◦ g)(α)
       α                 du                        g(a)        du
                                                                                                          b
                                        = F g(β) − F g(α) = F (b) − F (a) =                                   f (x) dx
                                                                                                      a


                                                                                                                         . . .

       1.4.2.                 6x2 (x3 + 2)99 dx

                                                                                         1
                    u = x3 + 2                     du = 3x2 dx                   dx =       du
                                                                                        3x2


                                                   1                                 2 100      1
    6x2 (x3 + 2)99 dx =            6x2 u99            du =             2u99 du =        u + c = (x3 + 2)100 + c
                                                  3x2                               100        50




       1.4.3.                 8e2x+3 dx
                                                                              16

                                                  1
         u = 2x + 3            du = 2dx       dx = du
                                                  2

                              1
         8e2x+3 dx =       8eu du =       4eu du = 4eu + c = 4e2x+3 + c
                              2


                  √
1.4.4.             3x + 4 dx

                                                  1
         u = 3x + 4            du = 3dx       dx = du
                                                  3

   √                      √ 1     1 u3/2      2 3       2         3
    3x + 4 dx =            u du =        + c = u 2 + c = (3x + 4) 2 + c
                            3     3 3/2       9         9



1.4.5.            x(5 + 3x2 )8 dx

                                                        1
         u = 5 + 3x2            du = 6xdx       dx =      du
                                                       6x


                               1          1 8     1        1
x(5 + 3x2 )8 dx =       xu8      du =       u du = u9 + c = (5 + 3x2 )9 + c
                              6x          6       54       54



                    √
1.4.6.            x2 1 + x dx

              √
         u=       1+x           u2 = 1 + x      x = u2 − 1      dx = 2udu




      √
    x2 1 + x dx =             (u2 − 1)2 u(2u)du =    (u4 − 2u2 + 1)(2u2 )du
                                                 2     4      2
                      =    2u6 − 4u4 + 2u2 du = u7 − u5 + u3 + c
                                                 7     5      3
                        2         7  4        5  2        3
                      =   (1 + x) 2 − (1 + x) 2 + (1 + x) 2 + c
                        7            5           3
                                                                                                                    17

                                              3    √
        1.4.7.                                    x 1 + x dx
                                          0

                                √
                       u=           1+x               u2 = 1 + x           x = u2 − 1            dx = 2udu       x=0
            √                                                         √
       u=    1+0=1                            x=3                u=       1+3=2



                       3    √                              2                           2
                           x 1 + x dx =                        (u2 − 1)u(2u)du =           (2u4 − 2u2 )du
                   0                                   1                           1
                                                                      2
                                                      2 5 2 3               64 16 2 2   116
                                                  =     u − u             =   −   − + =
                                                      5    3          1     5   3  5 3  15




1.4.2                                                               (integration by parts)

                 d(uv) = udv + vdu
uv =    d(uv) =                u dv +         v du



                                                      u dv = uv −          v du

               u                                                      f                                      v

                                     f



            1.4.8.                                                    (integration by parts)                 u(x)
v(x)                                 u (x)             v (x)

                           b                                                           b
                               u(x)v (x) dx = u(b)v(b) − u(a)v(a) −                        v(x)u (x) dx
                       a                                                           a


                                 h(x) = u(x)v(x)                       h (x) = u(x)v (x) + v(x)u (x)
                                                                                                                   18




                           b                        b
                               h (x) dx =               u(x)v (x) + v(x)u (x) dx
                       a                        a
                                                    b                        b
                       h(b) − h(a) =                    u(x)v (x) dx +           v(x)u (x) dx
                                                a                        a
                                                    b                        b
       u(b)v(b) − u(a)v(a) =                            u(x)v (x) dx +           v(x)u (x) dx
                                                a                        a
                  b                                                                  b
                      u(x)v (x) dx = u(b)v(b) − u(a)v(a) −                               v(x)u (x) dx
              a                                                                  a


                                                                                                                 . . .

   1.4.9.                      xex dx


             u=x                   dv = ex dx                  v=     dv =               ex dx = ex        du = dx




              xex dx = xex −                ex dx = xex − ex + c = (x − 1)ex + c




   1.4.10.                      ln x dx

                                                                                                                 1
              u = ln x                  dv = dx                 v=       dv =              dx = x         du =     dx
                                                                                                                 x




                                  1
 ln x dx = x ln x −              x dx = x ln x −             1 dx = x ln x − x + c = x(ln x − 1) + c
                                  x



                                    2
   1.4.11.                      x3 ex dx

                                            2                                               2             1 x2      1 2
             u = x2               dv = xex dx                 v=     dv =                xex dx =     d     e      = ex
                                                                                                          2         2
du = 2xdx
                                                                                         19




       2    1     2         1 x2        1     2        2    1     2 1 2     1         2
  x3 ex dx = x2 ex −          e (2x)dx = x2 ex −    xex dx = x2 ex − ex +c = (x2 −1)ex +c
            2               2           2                   2       2       2




       1.4.12.            x sin(2x)dx

                                                                                 1
                 u=x        dv = sin(2x)dx         v=    dv =       sin(2x)dx = − cos(2x)
                                                                                 2
    du = dx



                    1                     1             1           1
     x sin(2x)dx = − x cos(2x) −         − cos(2x)dx = − x cos(2x) + sin(2x) + c
                    2                     2             2           4




       1.4.13.            x2 e3x dx

                                                                                1
                   u = x2        dv = e3x dx            v=      dv =    e3x dx = e3x
                                                                                3
du = 2xdx


                            1             1 3x        1         2
                 x2 e3x dx = x2 e3x −       e (2x)dx = x2 e3x −        xe3x dx
                            3             3           3         3

                                      xe3x dx                       u=x          dv = e3x dx
         1
      v = e3x          du = dx
         3

                              1              1 3x    1      1
                     xe3x dx = xe3x −          e dx = xe3x − e3x + c1
                              3              3       3      9




                       1 2 3x     2            1         2 1 3x 1 3x
       x2 e3x dx =       xe −         xe3x dx = x2 e3x −      xe − e + c1
                       3          3            3         3 3      9
                       1 2 3x     2 3x     2                   2
                     =   xe −       xe + e3x + c         c = − c1
                       3          9       27                   3
                                                                                                   20




         1.4.14.               ex sin x dx


                   u = sin x         dv = ex dx         v=     dv =     ex dx = ex         du = cos x dx




                                 ex sin x dx = ex sin x −    ex cos x dx

                                          ex cos x dx                          u = cos x       dv =

ex dx         v = ex           du = − sin x dx



             ex cos x dx = ex cos x −         ex (− sin x) dx = ex cos x +     ex sin x dx




        ex sin x dx = ex sin x −          ex cos x dx = ex sin x − ex cos x +        ex sin x dx

        ex sin x dx = ex sin x − ex cos x −          ex sin x dx

  2     ex sin x dx = ex (sin x − cos x)
                          1 x
        ex sin x dx =       e (sin x − cos x) + c
                          2




1.4.3                                        (partial fractions method)
                                                                             P (x)
                                       (rational function)         f (x) =                 P       Q
                                                                             Q(x)
                      (polynomial functions)



                                    x−1
         1.4.15.                              dx
                               x3   − x2 − 2x
                                                                                   21


                x3 − x2 − 2x = x(x − 2)(x − 1)



     x−1        A  B   C    A(x + 1)(x − 2) + Bx(x + 1) + Cx(x − 2)
               = +   +    =
x3   −x 2 − 2x  x x−2 x+1               x(x − 2)(x + 1)

          (numerator)


                 x − 1 = A(x + 1)(x − 2) + Bx(x + 1) + Cx(x − 2)

                        = Ax2 − Ax − 2A + Bx2 + Bx + Cx2 − 2Cx

                        = (A + B + C)x2 + (−A + B − 2C)x + (−2A)


                                  (method of undetermined coefficient)   A+B+C = 0
                                                                           1     1
 −A+B−2C = 1                −2A = −1                                     A= ,B=
                                                                           2     6
     2
 C=−
     3


                        1             1             2
     x−1                2             6
                                                   −3     1       1         2
  3 − x2 − 2x
              dx =          dx+            dx+        dx = ln |x|+ ln |x−2|− ln |x+1|+c
x                       x          x−2            x+1     2       6         3



                         x3 + 1
      1.4.16.                   dx
                         x−1

                              (numerator)             (degree)               (denominator)



                                      x2    +    x    +   1
                            x−1       x3    +    1
                                      x3    −    x
                                                 x2   +   1
                                                 x2   −   x
                                                          x + 1
                                                          x − 1
                                                              2
                                                                                   22

        x3 + 1                 2
               = x2 + x + 1 +
        x−1                   x−1


      x3 + 1                            2      1    1
             dx =      x2 + x + 1 +        dx = x3 + x2 + x + 2 ln |x − 1| + c
      x−1                              x−1     3    2



                          x+1
     1.4.17.                      dx
                        x(x − 1)2




         x+1      A    B       C     A(x − 1)2 + Bx + Cx(x − 1)
                 = +        +      =
       x(x − 1)2  x (x − 1)2 x − 1            x(x − 1)2
         (numerator)


x + 1 = Ax2 − 2Ax + A + Bx + Cx2 − Cx = (A + C)x2 + (−2A + B − C)x + A


                              (method of undetermined coefficient)         A+C = 0
 −2A + B − C = 1             A=1                                        A = 1, B = 2
 C = −1



  x+1               1            2               −1                2
          dx =        dx +             dx +         dx = ln |x| −     − ln |x − 1| + c
x(x − 1)2           x         (x − 1)2          x−1               x−1



                        2x2 + 3x + 9
     1.4.18.                         dx
                          x3 − 27

               x3 − 27 = (x − 3)(x2 + 3x + 9)



   2x2 + 3x + 9     A     Bx + C     A(x2 + 3x + 9) + (Bx + C)(x − 3)
                =      + 2         =
     x3 − 27      x − 3 x + 3x + 9         (x − 3)(x2 + 3x + 9)
                                                                                                               23


             (numerator)


2x2 +3x+9 = Ax2 +3Ax+9A+Bx2 −3Bx+Cx−3C = (A+B)x2 +(3A−3B+C)x+(9A−3C)


                                       (method of undetermined coefficient)                         A+B = 2
                                                                                                         4
   3A − 3B + C = 3                9A − 3C = 9                                                         A= ,
                                                                                                         3
   2
B=       C =1
   3




                                  4                   2            4
    2x2 + 3x + 9                  3
                                                   +1 3
                                                        x          3        1                            2x + 3
                 dx =               dx +                 dx =          dx +                                       dx
      x3 − 27                 x−3           x 2 + 3x + 9         x−3        3                       x2   + 3x + 9
                           4              1         1
                         =   ln |x − 3| +                  d(x2 + 3x + 9)
                           3              3    x2 + 3x + 9
                           4              1
                         =   ln |x − 3| + ln |x2 + 3x + 9| + c
                           3              3




1.5                                                                       (Differentiation a Definite
        Integral)

             x                        (parameter)                                                   f (x, t)
t                    t = a(x)           t = b(x)               a          b                                x
                                                                      b(x)
                                                                              f (x, t) dt
                                                                     a(x)
                                                                    b(x)
                         x                            A(x) =              f (x, t) dt
                                                                   a(x)

            x                                                                                A(x)
                                                                              b(x)
                                                       dA        d
                                                          (x) =                      f (x, t) dt
                                                       dx       dx          a(x)
                                                                                                                ?
             b(x)                                   b(x)
        d                                                  ∂f
                    f (x, t) dt                               (x, t) dt               ?
       dx   a(x)                                a(x)       ∂x


                             ?
                                                                                                                                              24




                (Leibnitz’s Formula)

                                                  a(x) = a                      b(x) = b                                 (constant functions)
                                              1.5.2


                                                                                                               b
                                                                                            2       d
             1.5.1.        f (x, t)                                                 R                              f (x, t) dt = f (x, b)
                                                                                                    db     a
         b
 d
             f (x, t) dt = −f (x, a)
da   a


                                x                                               F (t) =           f (x, t) dt                                  f

(                     t             x         )

                                b
                           d                     d
                                    f (x, t) dt =   F (b) − F (a) = F (b) = f (x, b)
                          db   a                 db
                                 b
                           d                      d
                                   f (x, t) dt =    F (b) − F (a) = −F (a) − f (x, a)
                          da   a                 da

                                                                                                                                            . . .

                                              ∂f
             1.5.2.        f (x, t)              (x, t)                                                  [a, b] × R
                                              ∂x
                                                      b                                 b
                                           d                                                ∂f
                                                          f (x, t) dt =                        (x, t) dt
                                          dx      a                                 a       ∂x

                                                                                                                                      ∂f
                               h                                                            ε(h) = f (x + h, t) − f (x, t) − h           (x, t)
                                                                                                                                      ∂x
                                        ∂f              f (x + h, t) − f (x, t)                                                        ε(h)
                                           (x, t) = lim                                                                            lim       =0
                                        ∂x          h→0            h                                                               h→0 h
                                                              b
                                        A(x) =                    f (x, t) dt
                                                          a



                                          b                                     b  b
    A(x + h) − A(x)                      a
                                            f (x + h, t) dt − a f (x, t) dt          f (x + h, t) − f (x, t)
                    =                                                         =                              dt
           h                                              h                      a              h
                                           b                                b                     b
                                               ∂f           ε(h)              ∂f                    ε(h)
                                    =             (x, t) +          dt =         (x, t) dt +             dt
                                         a     ∂x            h            a ∂x                  a    h
                                           b
                                             ∂f              ε(h)
                                    =           (x, t) dt +       (b − a)
                                         a ∂x                 h
                                                                                                                                      25


                                h→0



         b                                                                               b
 d                           dA(x)          A(x + h) − A(x)                                  ∂f                 ε(h)
             f (x, t) dt =          = lim                   =                                   (x, t) dt + lim      (b − a)
dx   a                         dx      h→0         h                                 a       ∂x             h→0 h
                               b
                                 ∂f
                           =        (x, t) dt
                              a ∂x


                                                                                                                                    . . .




                                                                                                               ∂f
               1.5.3.                     (Leibnitz’s Formula)                      f (x, t)                      (x, t)
                                                                                                               ∂x
                 R2          a(x)           b(x)                                                                       (continuously
differentiable)



                 b(x)                                                                                         b(x)
          d                                             db(x)             da(x)                                      ∂f
                        f (x, t) dt = f x, b(x)               − f x, a(x)       +                                       (x, t) dt
         dx     a(x)                                     dx                dx                             a(x)       ∂x

                                          b(x)                                                     b
                             A(x) =              f (x, t) dt              I(a, b, x) =                 f (x, t) dt
                                       a(x)                                                    a

A(x) = I a(x), b(x), x                                                        (total differentiation)



         dA(x)   ∂I a(x), b(x), x da(x) ∂I a(x), b(x), x db(x) ∂I a(x), b(x), x
               =                       +                      +
          dx           ∂a          dx         ∂b          dx         ∂x

                1.5.1               1.5.2

                                                        b
               ∂I a(x), b(x), x             ∂
                                      =                     f (x, t) dt                      = −f x, a(x)
                     ∂a                     ∂a      a                      a(x),b(x),x
                                                        b
               ∂I a(x), b(x), x             ∂
                                      =                     f (x, t) dt                      = f x, b(x)
                     ∂b                     ∂b      a                      a(x),b(x),x
                                                        b                                               b(x)
               ∂I a(x), b(x), x             ∂                                                                  ∂f
                                      =                     f (x, t) dt                      =                    (x, t) dt
                     ∂x                     ∂x      a                       a(x),b(x),x                a(x)    ∂x
                                                                                                  26




      b(x)                                                                  b(x)
 d                           dA(x)             db(x)             da(x)             ∂f
             f (x, t) dt =         = f x, b(x)       − f x, a(x)       +              (x, t) dt
dx   a(x)                     dx                dx                dx       a(x)    ∂x

                                                                                                . . .

                                                                                            3
                             db(x)
               f x, b(x)
                              dx
                                                                da(x)
                                                   −f x, a(x)
                                                                 dx




                        a(x)
                                                                                     b(x)
                                                                                            ∂f
                        a(x)                                                                   (x, t) dt
                                                                                    a(x)    ∂x
                                                                             f (x, t)
                                                   1.3




                                     1.3:
                                                                                                       27




          1.5.4.                                                                   (1)                 (2)




                2
      d                     (1)  d        t=2    d
1.                  e−x dt =         te−x t=0 =      2e−x = −2e−x
     dx     0                   dx              dx
                                   2              2
                            (2)      ∂e−x                                        t=2
                            =             dt =      −e−x dt = −te−x              t=0
                                                                                       = −2e−x
                                 0    ∂x        0
                                        t=3                                                     1
    d           3
                     t   (1)d     xt              d x 3 − x2      (ln x)(3x2 − 2x) − (x3 − x2 ) x
2.                  x dt =                  =                  =
   dx       2              dx    ln x t=2 dx            ln x                   (ln x)2
                                   2             2
                                3x − 2x x − x
                             =              −
                                   ln x        (ln x)2
                              3                3                                      t=3
                         (2)    ∂xt                t−1       1 3 t         1      xt         3
                                                                                               xt
                         =           dt =        tx    dt =      tx dt =       t          −        dt
                             2 ∂x            2               x 2           x     ln x t=2   2 ln x
                                                 t=3                   t=3
                                    1     txt          1      xt                 3x2 − 2x x2 − x
                                  =                  −                       =           −
                                    x    ln x    t=2   x    (ln x)2    t=2         ln x    (ln x)2
                2x
      d                     (1)    d 3    t=2xd
3.                   3t2 dt =        t    t=0
                                                 = 8x3 = 24x2
     dx     0                     dx         dx
                                                 2x
                            (2)      d(2x)          ∂(3t2 )
                            = 3(2x)2       +                dt = 24x2
                                      dx       0     ∂x




                                                                                         +∞      −∞




                                            ∂f
           1.5.5.           f (x, t)           (x, t)                                   R2            p(t)
                                            ∂x                         ∞
                ∂f (x, t)
 p(t) ≥                                          (x, t) ∈ R2               p(t) dt < ∞
                  ∂x                                                  −∞

                                           ∞                     ∞
                                     d                                ∂f
                                                 f (x, t) dt =           (x, t) dt
                                    dx    −∞                     −∞   ∂x
                                                                                                    28


                                                                                              (Lebesgue
Dominated Convergence Theorem)




                                                                                                  . . .


1.6                                                  (Multiple Riemann Integral)

                             (Riemann Integration)
        [a, b]                                           (area)                                 f (x, y)

      f (x, y, z)


                                                                                         (Double Riemann

Integration)                                                                                  (volume)


          (Multiple Riemann Integraion)

                 D ⊂ [a, b] × [c, d] ⊂ R2                                          f :D→R


                     (partition)       D             n            E1 , E2 , . . . , En

1.4




                            1.4:
                                                                                                                           29


                                                        (diameter)                                                     0
                                         (n → ∞)

    diam(Ei )               i = 1, 2, . . . , n              lim    max diam(Ei )      =0                   µ(Ei )
                                                            n→∞     1≤i≤n

                Ei                                           diam(Ei ) → 0            µ(Ei ) → 0             (
                        )
                                 i = 1, 2, . . . , n                  fi                  (infimum)               f (x, y)

               Ei                     fi =       inf    f (x, y)
                                             (x,y)∈Ei
                                                                                                        n
                                                    (Lower Riemann Sum)                      Sn =           f i µ(Ei )
                                                                                                    i=1
                                           i = 1, 2, . . . , n              fi                    (supremum)
f (x, y)                    Ei                   f i = sup f (x, y)
                                                         (x,y)∈Ei
                                                                                                                 n
                                                                (Upper Riemann Sum)                 Sn =               f i µ(Ei )
                                                                                                                 i=1
                    V                                   f (x)               D                Sn ≤ V ≤ Sn
                                           D                                 n



                                                         S 1 ≤ S 2 ≤ . . . S n−1 ≤ S n ≤ S n+1 ≤ . . .
(S n )∞
      n=1                                (nondecreasing sequence)                             (upper bound)

    V                   (S n )∞
                              n=1

(Lower Double Riemann Integral)                             f (x, y) d(x, y) = lim S n              V
                                                                                n→∞
                                                        D

                                                 f (x, y) d(x, y) ≤ V
                                             D
                                                                    D                         n



                                                                     S 1 ≥ S 2 ≥ . . . S n−1 ≥ S n ≥ S n+1 ≥ . . .
        (S n )∞
              n=1                                  (nonincreasing sequence)                         (lower bound)
           V                         f (x, y)                                D              S1 < ∞

(S n )∞
      n=1                                                                                          (Upper Double
Riemann Integral)                    f (x, y) d(x, y) = lim S n                    V
                                 D                            n→∞
                                                                                                                                30


                              f (x, y) d(x, y) ≥ V
                          D

                                                                          f (x, y) d(x, y) ≤ V ≤                     f (x, y) d(x, y)
                                                                      D                                          D

                               f (x, y) d(x, y) =                  f (x, y) d(x, y)
                           D                                   D

       f (x, y) d(x, y) = V =                    f (x, y) d(x, y)
   D                                         D

(Double Riemann Integral) V =                          f (x, y) d(x, y)
                                                   D




               1.6.1.         D = [a, b] × [c, d]               f :D→R




                                             b         d                                  d       b
                   f (x, y) d(x, y) =                      f (x, y) dy        dx =                    f (x, y) dx       dy
               D                         a         c                                  c       a


                                     [a, b] × [c, d]                                 [a, b]                  n                a =
x0 < x1 < x2 < · · · < xn−1 < xn = b
       0                                                    (n → ∞)

      ∆xi = xi − xi−1                    i = 1, 2, . . . , n                  lim     max ∆xi            =0
                                                                              n→∞    1≤i≤n

      [c, d]                  m                  c = y0 < y1 < y2 < · · · < ym−1 < ym = d
                                                                     0                                                  (m → ∞)

                                                                ∆yj = yj − yj−1                         j = 1, 2, . . . , m
lim        max ∆yj         =0                       [a, b]           [c, d]                                          [a, b] × [c, d]
m→∞     1≤j≤m

                        nm                                    i = 1, 2, . . . , n         j = 1, 2, . . . , m

           Eij = [xi−1 , xi ] × [yj−1 , yj ]                                  Eij             µ(Eij ) = ∆xi ∆yj
                                             1.5
                                                                                                                      31




             1.5:                                                                              [a, b] × [c, d]



       f ij =     inf          f (x, y)         f ij =     sup f (x, y)                              x ∈ [a, b]
                (x,y)∈Eij                                (x,y)∈Eij
f j (x) =       inf           f (x, y)          f j (x) =      sup        f (x, y)                          y ∈ [c, d]
            yj−1 ≤y≤yj                                      yj−1 ≤y≤yj

         f i (y) =            inf    f (x, y)        f i (y) =         sup        f (x, y)
                      xi−1 ≤x≤xi                                     xi−1 ≤x≤xi
                          d
         g(x) =               f (x, y) dy
                      c
            f


                                                                          n   m                                  n   m
    f (x, y)d(x, y) =                                        lim
                                          f (x, y)d(x, y) = n→∞                      f ij µ(Eij ) = lim lim                f ij µ(Eij )
D                                                                                                n→∞ m→∞
                                     D                          m→∞ i=1 j=1                                      i=1 j=1
                                                                     n   m                                        n   m
                          =                                  lim
                                          f (x, y)d(x, y) = n→∞                      f ij µ(Eij ) = lim lim                f ij µ(Eij )
                                     D                                                           n→∞ m→∞
                                                                m→∞ i=1 j=1                                      i=1 j=1


                                                                      g
                                                                                                              32




b                     n                                 n
    g(x) dx =   lim         g i ∆xi = lim                            inf       g(x) ∆xi
                n→∞                      n→∞                      xi−1 ≤x≤xi
a                     i=1                               i=1
                       n                            d
           =    lim            inf                      f (x, y) dy ∆xi
                n→∞         xi−1 ≤x≤xi          c
                      i=1
                       n                                      m
           =    lim            inf             lim                f j (x)∆yj     ∆xi
                n→∞         xi−1 ≤x≤xi         m→∞
                      i=1                                 j=1
                       n                                 m
           ≥    lim         lim          inf                  f j (x)∆yj ∆xi
                n→∞         m→∞ xi−1 ≤x≤xi
                      i=1                               j=1
                       n             m
           ≥    lim         lim                inf            f j (x)∆yj ∆xi
                n→∞         m→∞           xi−1 ≤x≤xi
                      i=1          j=1
                               n    m
           =    lim lim                        inf                 inf     f (x, y)∆yj ∆xi
                n→∞ m→∞                   xi−1 ≤x≤xi yj−1 ≤y≤yj
                              i=1 j=1
                               n   m
           =    lim lim                        inf          f (x, y)µ(Eij ) =              f (x, y) d(x, y)
                n→∞ m→∞                   (x,y)∈Eij                                    D
                              i=1 j=1


                                               g


b                     n                                 n
    g(x) dx =   lim         g i ∆xi = lim                           sup        g(x) ∆xi
a               n→∞                      n→∞                      xi−1 ≤x≤xi
                      i=1                               i=1
                       n                            d
           =    lim           sup                       f (x, y) dy ∆xi
                n→∞         xi−1 ≤x≤xi          c
                      i=1
                       n                                      m
           =    lim           sup              lim                f j (x)∆yj     ∆xi
                n→∞         xi−1 ≤x≤xi         m→∞
                      i=1                                 j=1
                       n                                 m
           ≤    lim         lim          sup                  f j (x)∆yj ∆xi
                n→∞         m→∞ xi−1 ≤x≤xi
                      i=1                               j=1
                       n             m
           ≤    lim         lim                sup            f j (x)∆yj ∆xi
                n→∞         m→∞           xi−1 ≤x≤xi
                      i=1          j=1
                               n    m
           =    lim lim                        sup                 sup     f (x, y)∆yj ∆xi
                n→∞ m→∞                   xi−1 ≤x≤xi yj−1 ≤y≤yj
                              i=1 j=1
                               n   m
           =    lim lim                        sup f (x, y)µ(Eij ) =                       f (x, y) d(x, y)
                n→∞ m→∞
                              i=1 j=1 (x,y)∈Eij                                        D
                                                                                                                                        33

                               b                                                                 b
                                   g(x) dx ≥                        f (x, y) d(x, y) ≥               g(x) dx
                               a                                D                                a


  b                   b
      g(x) dx =           g(x) dx =               f (x, y) d(x, y)                         g
  a                   a                       D
                                          b                                b       d
               f (x, y) d(x, y) =             g(x) dx =                                f (x, y) dy        dx
           D                          a                                a       c
                                                                                                                               b
                                                                                                                  h(y) =           f (x, y) dx
                                                                                                                           a




                                                                                                                                     . . .


            1.6.2.                                (2x + 3y + 4)d(x, y)                              D = [0, 1] × [0, 2]
                                              D




                                                       2        1                                         2
                                                                                                                               x=1
           (2x + 3y + 4)d(x, y) =                                   (2x + 3y + 4)dxdy =                       x2 + 3xy + 4x    x=0
                                                                                                                                     dy
       D                                           0        0                                         0
                                                       2
                                                                            3                   2
                                      =                    (5 + 3y)dy = 5y + y 2                     = 16
                                                   0                        2                   0




                     D                                              (unbounded set)                                           (Improper
Integral)                                                                                  D1 ⊂ D2 ⊂ . . .
                            (increasing sequence of bounded sets)                                                lim Dn = D
                                                                                                                 n→∞

                                                           (Improper Double Riemann Integration)
       f (x, y) d(x, y) = lim                     f (x, y) d(x, y)
   D                          n→∞         Dn

        (Dn )∞
             n=1

                                                                                                     D




                                                                                                                  (Fubini Theorem)
                                                                                                                       34


              1.6.3.                     (Fubini Theorem)                               |f (x, y)|d(x, y) < ∞
                                                                                   R2



                                    ∞          ∞                                        ∞      ∞
              f (x, y) d(x, y) =                   f (x, y) dy           dx =                      f (x, y) dx    dy
         R2                        −∞         −∞                                    −∞        −∞


                                                                                                                  (Tonelli
Theorem)                                                    (Monotone Convergence Theorem)




                                                                                                                    . . .


          1.6.4.                             e−(x+y) d(x, y)
                                        R2
                                         +




                                                   ∞        ∞                           ∞
                                                                                                      x→∞
                       e−(x+y) d(x, y) =                        e−x−y dxdy =                 −e−x−y   x=0
                                                                                                             dy
                 R2
                  +                            0        0                           0
                                                   ∞                           ∞
                                        =              e−y dy = −e−y               =1
                                               0                               0




1.6.1                                              (Change of Variables Technique)




                   (Change of Variables Technique)
                                   (x, y)                                          (u, v)                (x, y) = g(u, v)
     g                               (one–to–one function)
                                    d(x, y)                                                        d(x, y)
                   d(x, y) = det            d(u, v)                                         J =
                                    d(u, v)                                                        d(u, v)
               (Jacobian)

                                                                    ∂x    ∂x
                                                d(x, y)             ∂u    ∂v
                                        J=              =           ∂y    ∂y
                                                d(u, v)             ∂u    ∂v
                                                                                                      35




              1.6.5. (                        (change of variables))        f : D → R

               D         (x, y) = g(u, v)              g
J                                                      g(D ) = D



                              f (x, y) d(x, y) =           f g(u, v) | det J| d(u, v)
                          D                            D


                                                                                        (multidimensional
real analysis)


                                                                                                    . . .


          :                                                       det J                  (absolute value)




         1.6.6.                       (x2 + y 2 − 1)d(x, y)         D
                                  D
              x − y = −1, x − y = 1, x + y = 1                 x+y =5



                                   1.6




                                   1.6:            D
                                                                                                                            36




                                       1           x+1                                     2    x+1
       (x2 + y 2 − 1)d(x, y) =                           (x2 + y 2 − 1)dydx +                           (x2 + y 2 − 1)dydx
   D                               0            1−x                                    1       x−1
                                               3    5−x
                                 +                           (x2 + y 2 − 1)dydx
                                           2           x−1




                                                          1  1                       1  1
u=x−y             v =x+y                                x= u+ v                  y =− u+ v
                                                          2  2                       2  2

                                       ∂x          ∂x            1     1
                                       ∂u          ∂v            2     2
                                                                               1 1  1
                         det J =       ∂y          ∂y       =    1     1
                                                                           =    + =
                                       ∂u          ∂v
                                                                −2     2
                                                                               4 4  2

                     D                                               x − y = −1, x − y = 1, x + y = 1

x+y = 5                    u = −1, u = 1, v = 1                                v=5
                                 D = [−1, 1] × [1, 5]



                                                                       2                           2
          2   2                                         1   1                 1  1
       (x + y − 1)d(x, y) =                               u+ v             + − u+ v                    − 1 |det J| d(u, v)
   D                                   D                2   2                 2  2
                                       5           1
                                                            1 2 1 2             1
                            =                                 u + v −1            dudv
                                   1            −1          2    2              2
                                               5                               u=1
                              1                        1 3 1 2
                            =                            u + uv − u                   dv
                              2            1           6    2                  u=−1
                                               5                                 3             5
                              1                         2  5              1    v  5                     52
                            =                          v −           dv =        − v               =
                              2            1               3              2    3  3            1         3



                                                                                                                       (coordinate
system)                    (rectangular coordinate system) (x, y)                                      (x, y, z)
                                                                                                                             3

                            (Polar Coordinate System )                                                             (Cylindrical
Coordinate System)                                              (Spherical Coordinate System)
                                                                                                    37


                   (Polar Coordinate System )

                    (polar coordinate system)                                  (origin)
(reference line)                                               2

                                2                                      r
θ                          (radian)                  1.7 (a)




                                        1.7:




                          (arc length/radius)                              π
                                                        (circumference/diameter)
                                           πD        2πR           R                                D
                                                                                          2πR
                                                                                              = 2π
                                                                                           R


                      x                                                                   1.7 (b)

                                                x = r cos θ        y = r sin θ
                                                                                                                           38




                   ∂x   ∂x
                                    cos θ −r sin θ
     det J =       ∂r
                   ∂y
                        ∂θ
                        ∂y      =                  = r cos2 θ + r sin2 θ = r(cosθ + sin2 θ) = r
                   ∂r   ∂θ
                                    sin θ r cos θ

                                                        d(x, y) = r d(r, θ)                           1.8




                                              1.8:




          1.6.7.                                                  R        πR2

                        R             A = {(x, y) | x2 + y 2 ≤ R2 }
           A = {(r, θ) | 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π}



                                                   2π       R                  2π
                                                                                    r2   r=R     1       θ=2π
         d(x, y) =              r d(r, θ) =                     r drdθ =                     dθ = R2 θ             = πR2
     A                      A                  0        0                  0        2    r=0     2       θ=0




                                (Cylindrical Coordinate System)

                                                                                               (Cylindrical Coordinate
System)                                                                                                        x
                                                                                                                       1.9
                                                                                               39




                                        1.9:



                       (x, y, z)                                            (r, θ, h)

x = r cos θ, y = r sin θ      z=h


                                   ∂x     ∂x     ∂x
                                   ∂r      ∂θ    ∂h
                                                        cos θ −r sin θ 0
                                   ∂y     ∂y     ∂y
                     det J =       ∂r      ∂θ    ∂h
                                                      = sin θ r cos θ 0
                                   ∂z     ∂z     ∂z
                                   ∂r      ∂θ    ∂h
                                                          0      0     1
                                         2
                             = r cos θ + r sin2 θ = r(cosθ + sin2 θ) = r


                                                d(x, y, z) = r d(r, θ, h)               1.10




                              1.10:
                                                                                                                             40


            1.6.8.                                                                  R          H        πR2 H

                              R           H                   V = {(x, y, z) | x2 + y 2 ≤ R2 , 0 ≤ z ≤ H}
                                                                V = {(r, θ, h) | 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, 0 ≤

h ≤ H}



                                                                      H       2π        R
           d(x, y, z) =                       r d(r, θ, h) =                                r drdθdh
       V                              V                           0       0         0
                                  H           2π        r=R                H                 θ=2π
                                                   r2                              1 2                           h=H
                          =                                   dθdh =                 R θ            dh = πR2 h         = πR2 H
                              0           0        2    r=0               0        2         θ=0                 h=0




                          (Spherical Coordinate System)



                    (Spherical Coordinate System)

                r                 (projection)                         xy                          x    φ
           xy        θ1
      1.11




                                                        1.11:


  1
                                  θ                                   z
                                                                                             41


                         (x, y, z)                                       (r, φ, θ)          x=
r cos θ cos φ, y = r cos θ sin φ         z = r sin θ


               ∂x   ∂x    ∂x
               ∂r   ∂φ    ∂θ      cos θ cos φ −r cos θ sin φ −r sin θ cos φ
               ∂y   ∂y    ∂y
  det J =      ∂r   ∂φ    ∂θ
                                = cos θ sin φ r cos θ cos φ −r sin θ sin φ
               ∂z   ∂z    ∂z
               ∂r   ∂φ    ∂θ
                                     sin θ          0           r cos θ
               2    3       2
          = r cos θ cos φ + r2 cos θ sin2 θ sin2 φ + r2 cos θ sin2 θ cos2 φ + r2 cos3 θ sin2 φ

          = r2 cos3 θ + r2 cos θ sin2 θ = r2 cos θ(cos2 θ + sin2 θ) = r2 cos θ


                                              d(x, y, z) = r2 cos θ d(r, φ, θ)            1.12




                                     1.12:




                                                                  4 3
         1.6.9.                                           R         πR
                                                                  3

                     R       V = {(x, y, z) | x2 + y 2 + z 2 ≤ R2 }
                                                                 π  π
                    V = (r, φ, θ) | 0 ≤ r ≤ R, 0 ≤ φ ≤ 2π, − ≤ θ ≤
                                                                  2 2
                                                                                                                                 42



                                                                           π
                                                                           2
                                                                                   2π        R
                     d(x, y, z) =                   rd(r, φ, θ) =                                r2 cos θ drdφdθ
                 V                          V                             −π
                                                                           2
                                                                               0         0
                                        π                           r=R                       π                     φ=2π
                                                    2π
                                        2                r3                                   2   1 3
                                 =                          cos θ         dφdθ =                    R φ cos θ              dθ
                                      −π
                                       2
                                                0        3          r=0                      −π
                                                                                              2
                                                                                                  3                 φ=0
                                                           θ= π
                                      2 3                     2      2       2                          4
                                 =      πR sin θ                    = πR3 − − πR3                      = πR3
                                      3                    θ=− π     3       3                          3
                                                               2




1.7                                  (Special Functions)

                                                                                        (closed form)



                             2                                                 (Gamma Function)
(Beta Function)


1.7.1                                (Gamma Function)
                                                                                                               ∞
        1.7.1.                α>0                                                            Γ(α) =                xα−1 e−x dx
                                                                                                           0


                                                         α >0                                       α ≤0
                                            +∞                                                         α
(α ∈ Z+ )

                                                                                        1.13               1.7.2
                            1.7.3
                                                                                                                                            43




                                                                   1.13:


                                                                                                                      ∞               √
                                                                                                                            2
         1.7.2.                                                (Gaussian Integral Formula)                                e−x dx =     π
                                                                                                                     −∞
                                                  ∞                          ∞
                                                           2                          2
                                 A=                   e−x dx =                    e−y dy
                                              −∞                             −∞

(x, y)                                            (r, θ)                                                      R2



                             ∞                                 ∞                          ∞    ∞
                                          2                              2                                2 +y 2 )
         A2 =                     e−x dx                           e−y dy         =                e−(x              dxdy
                             −∞                            −∞                             −∞ −∞
                            2π ∞                                        2π                  ∞              2π
                                          −r2                                 1 2                               1     1         2π
                =                     e         rdrdθ =                      − e−r             dθ =               dθ = θ             =π
                        0     0                                     0         2            0          0         2     2         0

                                              ∞                     √
                                                       2
                             A=                   e−x dx =               π
                                          −∞

                                                                                                                                          . . .

            1.7.3.


   1. Γ(α + 1) = αΓ(α)

   2. Γ(n) = (n − 1)!                      n                                        (n ∈ Z+ )

            1           √
   3. Γ           =      π
            2
                        1             α−1
                                  1
   4. Γ(α) =                ln                    dx
                    0             x
                                                                                                                                 44




1.                                                                               u = xα                  dv = e−x dx

     v=    dv =         e−x dx = −e−x                        du = αxα−1 dx



                             ∞                                    ∞                                  ∞             ∞
     Γ(α + 1) =                  x(α+1)−1 e−x dx =                    xα e−x dx = −xα e−x                 −            αxα−1 −e−x dx
                         0                                    0                                      0+        0
                                                                                         ∞
                =       lim −xe−x − lim −xe−x +                                              αxα−1 e−x dx = αΓ(α)
                        x→∞            +             x→0                             0


                                                               lim −xe−x = 0
                                                              x→∞

                                             ˆ
                                         (l’Hopital’s rule)

2.                                                                        (Mathematical Induction)
                                             (                    n = 1)

                    ∞                                ∞                           ∞
       Γ(1) =           x1−1 e−x dx =                    e−x dx = −e−x                   = 0 − (−1) = 1 = 0! = (1 − 1)!
                0                                0                               0



                                             (induction step)                                                                k ∈ Z+
                                 k ∈ Z+                      Γ(k) = (k − 1)!

                         k+1                                                                                             1
     Γ(k + 1) = kΓ(k) = k(k − 1)! = k! = (k + 1) − 1 !                                                                        k+1


                                                                      ∞
                                                         1                   1
3.                                               Γ           =            t− 2 e−t dt
                                                         2        0
                        √                                2
                x=           t               t=x                      dt = 2xdx                      t=0                x=0

          t→∞                x→∞


                                         ∞                            ∞                                  ∞
                         1                       1                        1 −x2                                    2
                  Γ              =           t− 2 e−t dt =                  e (2x)dx = 2                     e−x dx
                         2           0                            0       x                          0

                                                                                                      2
                                                 (Gaussian Function) f (x) = e−x                                              (even
                                                                                                                                                45


       function)


                              ∞                         0                               ∞                             ∞
                                   −x2                           −x2                            −x2                            2
                                   e       dx =              e               dx +           e         dx = 2              e−x dx
                              −∞                        −∞                          0                             0


                                           1.7.2

                                                                 ∞                          ∞                      √
                                               1                              2                           2
                                       Γ               =2                e−x dx =                 e−x dx =             π
                                               2             0                              −∞


                                                                                         1
  4.                                                                              t = ln                       x = e−t                 dx = −e−t dt
                                                                                         x
             x → 0+                t→∞                           x=1                    t=0


                          1                α−1                   0                                        ∞
                                 1
                              ln                   dx =              tα−1 (−e−t )dt =                         tα−1 e−t dt = Γ(α)
                      0          x                           ∞                                        0


                                                                                                                                              . . .

                                                   5
          1.7.4.                           Γ
                                                   2

                                           1                3                                   1.7.3


              5               3                 3           3             3         1                     3 1              1           3√
         Γ         =Γ           +1             = Γ                       = Γ          +1              =    × Γ                     =      π
              2               2                 2           2             2         2                     2 2              2           4




1.7.2                             (Beta Function)

        1.7.5.                    α>0                   β>0

                                                                         1
                                           B(α, β) =                         xα−1 (1 − x)β−1 dx
                                                                     0


                                                        1.14
                                                                                             46




                                       1.14:


                       Γ(α)Γ(β)
  1.7.6. B(α, β) =
                       Γ(α + β)

                       B(α, β)Γ(α + β) = Γ(α)Γ(β)




                                   1                              ∞
   B(α, β)Γ(α + β) =                   xα−1 (1 − x)β−1 dx             y α+β−1 e−y dy
                                   0                          0
                                  1 ∞
                                                            β−1
                        =               (xy)α−1 (1 − x)y          ye−y dydx
                              0    0


                                                            u = xy            v = (1 − x)y
                      u
 y =u+v           x=
                     u+v

             ∂x   ∂x          v           −u
             ∂u   ∂v        (u+v)2      (u+v)2          v       −u        1
   det J =   ∂y   ∂y   =                         =           −         =
             ∂u   ∂v          1           1          (u + v)2 (u + v)2   u+v

0< x<1        0< y < ∞                   0 < u = xy < ∞               0 < v = (1 − x)y < ∞
                                                                                                                47




                                       1       ∞
                                                                          β−1
        B(α, β)Γ(α + β) =                          (xy)α−1 (1 − x)y             ye−y dydx
                                   0       0
                                       ∞        ∞
                                                                                    1
                              =                     uα−1 v β−1 (u + v)e−(u+v)          dvdu
                                   0        0                                      u+v
                                           ∞                          ∞
                              =                 uα−1 e−u du               v β−1 e−v dv    = Γ(α)Γ(β)
                                       0                          0


                                                                                                              . . .

                                       1
        1.7.7.                B 2,
                                       2

                     1.7.6

                                                     1          √     √
                          1         Γ(2)Γ            2     (1!)( π)     π  4
                     B 2,         =                  1
                                                         =      5
                                                                    = 3√ =
                          2         Γ 2+             2
                                                             Γ 2      4
                                                                        π  3




1.8                                                                                      (Economics and
       Econometrics Application)

                                                                                               (total)
                             (marginal)

        1.8.1.                                      (marginal cost function)                             q
                       C (q) = 3q 2 + 2q + 4                  /                             (fixed cost)
100                                 (total cost function)

      C(q) =     C (q) dq =       3q 2 + 2q + 4 dq = q 3 + q 2 + 4q + c                                      C(0) =

100              c = 100                                                    C(q) = q 3 + q 2 + 4q + 100




(Consumer Surplus)                                       (Producer Surplus)
                                                                                              48


(competitive market)                                                   1.15




                       1.15:



             QD (P )                             (demand function) QS (P )
(supply function) PD (Q)                                   (inverse demand function) PS (Q)
                        (inverse supply function)              P               (price) Q

(quantity)     P∗         Q∗                          (equilibrium price)                     (e
quilibrium quantity)                                                    (CS)
(P S)



                                    Q∗                             ∞
                    CS =                  PD (Q) − P ∗ dQ =            QD (P ) dP
                                0                               P∗
                                    Q∗                           P∗
                    PS =                  P ∗ − PS (Q) dQ =            QS (P ) dP
                                0                              0


         :                               (monopoly)

                                                              Deadweight Loss (DW L)
                                                                                                                               49


        1.8.2.                                                                         PD (Q) = 50 − 0.1Q
                         PS (Q) = 0.2Q + 20



                                                                    50 − 0.1Q∗ = 0.2Q∗ + 20                     Q∗ = 100

                                                (                                  )        P ∗ = 50−(0.1×100) = 40




                    Q∗                                       100                                     100
   CS =                  PD (Q) − P ∗ dQ =                         50 − 0.1Q − 40 dQ =                     10 − 0.1Q dQ
                0                                        0                                       0
                                       100
         = 10Q − 0.05Q2                      = 500
                                       0
                    Q∗                                       100                                      100
   PS =                  P ∗ − PS (Q) dQ =                         40 − (0.2Q + 20) dq =                    20 − 0.2Q dQ
                0                                        0                                        0
                                   100
         = 20Q − 0.1Q2                     = 1000
                                   0




                                                     X              Y                                 (continuous random
variables) P                                             (probability function)               E                         (event)




       1.8.3.               f (x)                                                                                 (probability

density function)            pdf             P (X ∈ E) =                f (x) dx
                                                                    E


          1.8.4.              f (x)                  pdf                  f (x) ≥ 0                         x           f (x) dx = 1
                                                                                                                    R


        1.8.5.                                                           pdf                                               1

  1.                                           (exponential distribution)

                                              1 x
                                       f (x) = e− θ                      x ∈ R+            θ>0
                                              θ
                                                                                                                50


2.                                    (standard normal distribution)

                                               1  z2
                                      f (z) = √ e− 2                           z∈R
                                               2π

3.                         (Gamma distribution)

                            1            x
               f (x) =         α
                                 xα−1 e− β                      x ∈ R+               α>0            β>0
                         Γ(α)β

4.                      (Beta distribution)

                       1
       f (x) =               xα−1 (1 − x)β−1                      x ∈ [0, 1]               α>0         β>0
                     B(α, β)




                     x
1.             u=       x = θu                      dx = θ du                          x=0           u=0
                     θ
     x→∞            u→∞

                                 ∞                  ∞                                  ∞
                                     1 −x               1 −u
                                       e θ =              e θ du = −e−θ                    =1
                             0       θ          0       θ                              0


               z                       √                          √
2.          u= √                 z=        2u           dz =          2 du                 z → −∞         u → −∞
                 2
           z→∞               u→∞                                                                (      1.7.2)


                    ∞
                         1  z2       1                      ∞
                                                                   2   √           1 √ √
                        √ e− 2 dz = √                           e−u        2 du = √   2 π=1
                   −∞    2π           2π                −∞                         2π
                     x
3.             u=       x = βu                      dx = β du                          x=0           u=0
                     β
     x→∞            u→∞



               ∞                                            ∞
                      1            x                               1
                         α
                           xα−1 e− β dx =                               (βu)α−1 e−u β du
           0       Γ(α)β                                0       Γ(α)β α
                                                                               ∞
                                                       1                                            Γ(α)
                                                =           βα                     uα−1 e−u du =         =1
                                                    Γ(α)β α                0                        Γ(α)
                                                                                                                                   51

             1                                                              1
                   1                            1                                                                  B(α, β)
    4.                   xα−1 (1 − x)β−1 dx =                                   xα−1 (1 − x)β−1 dx =                       =1
         0       B(α, β)                      B(α, β)                   0                                          B(α, β)




         1.8.6.                f (x, y)                                                                                        (joint

probability density function)                      joint pdf        P (X, Y ) ∈ E =                             f (x, y) d(x, y)
                                                                                                            E


                 1.8.7.           f (x, y)                 joint pdf                        f (x, y) ≥ 0                      x

y                     f (x, y) d(x, y) = 1
                 R2

             1.8.8.                f (x, y) = 1             0<x<1                         x < y < x+1                   f (x, y) = 0
    (x, y)                                     f (x, y)        joint pdf                           P (X + Y ≤ 1)

                                      f (x, y) ≥ 0                              (x, y)                                  f (x, y)
joint pdf                                                                       1

                  D = {(x, y) | 0 < x < 1, x < y < x + 1}



                                           1       x+1                  1           y=x+1              1               x=1
                 f (x, y) d(x, y) =                      1 dydx =           y               dx =           1 dx = x          =1
          D                            0       x                    0               y=x            0                   x=0




E{(x, y) | 0 < x < 1, x < y < x + 1, x + y ≤ 1}                                             1.16




                                           1.16:               D                                   E
                                                                                                                                               52




                                                                                      1                                   1
                                                                                      2
                                                                                                1−x                       2       y=1−x
        P (X + Y ≤ 1) =                          f (x, y) d(x, y) =                                   1 dydx =                y           dx
                                             E                                    0         x                         0           y=x
                                             1
                                             2                                            x= 1
                                                                                             2        1 1  1
                                                                                  2
                                =                (1 − 2x) dx = x − x                             =     − =
                                         0                                                x=0         2 4  4




        1.8.9.        X                                                     pdf           f (x)                                    (Expectation)
     g(x)                     Eg(X) =                 g(x)f (x) dx                        E|g(X)| < ∞
                                                  R




              1.8.10.                    X                                            g               h
          a

  1. E g(X) + h(X) = Eg(X) + Eh(X)

  2. E ag(X) = aEg(X)

        1.8.11.                 (mean)                                                    X           EX =            xf (x) dx
                                                                                                                  R

          1.8.12.                                                       X
θ>0

        pdf                                                                                      θ > 0
                                                                                                                      1 x
1.8.5                                                                                           u = x             dv = e− θ dx
                                                                                                                      θ
                      1 −x           x
v=      dv =            e θ dx = −e− θ                            du = dx
                      θ


                                                          ∞                                                           ∞
                                                                   1 −x                               x   ∞                        x
         EX =                 xf (x) dx =                     x      e θ      dx = −xe− θ                 0
                                                                                                              −           −e− θ dx
                          R                           0            θ                                              0
                           ∞
                                    −x                        −x    ∞
                  =             e    θ   dx = −θe              θ
                                                                    0
                                                                        =θ
                          0
                                                                                                                                   53

                      x
          lim xe− θ = 0                                               ˆ
                                                                  (l’Hopital’s rule)
          x→∞




        1.8.13.                     (variance)                                             X        V ar X = E (X − EX)2

          1.8.14.                                                           X
             θ>0

                          1.8.12
θ>0         EX = θ
                       1 x                                                  x
   u = (x − θ)2   dv = e− θ dx                                v = −e− θ                  du = 2(x − θ)dx
                       θ
                         1 x                                                         x
            u=x−θ    dv = e− θ dx                                  v = −e− θ                    du = dx
                         θ


                                                ∞
                                                                   1 −x
 V ar X = E (X − EX)2 =                             (x − θ)2         e θ         dx
                                            0                      θ
                                                ∞                                                       ∞
                                x   ∞                                       x                                             x
            = −(x − θ)2 e− θ        0
                                        −             −2(x − θ)e− θ dx = θ2 + 2                             (x − θ)e− θ dx
                                            0                                                       0
                                                              ∞                                                      ∞
                                        −x      ∞                      −x                                                     x
            = θ2 + 2       −(x − θ)e     θ
                                                0
                                                     −            −e    θ       dx       = θ2 + 2 −θ +                   e− θ dx
                                                          0                                                      0
                                         x ∞
                  2
            = θ + 2 −θ +            −θe− θ 0                  2
                                                         = θ + 2(−θ + θ) = θ                2



                            x                                        x
          lim (x − θ)2 e− θ = 0                 lim (x − θ)e− θ = 0                                                      ˆ
                                                                                                                     (l’Hopital’s
          x→∞                               x→∞

rule)

				
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posted:11/7/2011
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