422 2 Lecture Notes for EC422 Mathematical Economics 2 1 _Integral by xiaopangnv

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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

1.7:

(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)

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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