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					                                                        Table of Integrals∗
                             Basic Forms                                                          Integrals with Roots

                                     1                                                           √               2
                          xn dx =       xn+1 + c                    (1)                              x − adx =     (x − a)3/2 + C             (17)
                                    n+1                                                                          3
                              1                                                                         1        √
                                dx = ln x + c                       (2)                            √       dx = 2 x ± a + C                   (18)
                              x                                                                        x±a
                                                                                                       1         √
                             udv = uv −     vdu                     (3)                           √       dx = −2 a − x + C                   (19)
                                                                                                      a−x
                       1        1
                            dx = ln |ax + b| + c                    (4)
                     ax + b     a                                                  √         2             2
                                                                                  x x − adx = a(x − a)3/2 + (x − a)5/2 + C                    (20)
                                                                                             3             5
              Integrals of Rational Functions
                                                                                         √                   2b   2x     √
                            1             1                                                  ax + bdx =         +            ax + b + C       (21)
                                  dx = −     +c                     (5)                                      3a    3
                         (x + a)2        x+a
                                                                                                                  2
                            (x + a)n+1                                                     (ax + b)3/2 dx =         (ax + b)5/2 + C           (22)
                         n
               (x + a) dx =            + c, n = −1                  (6)                                          5a
                               n+1
                                                                                                  x      2        √
                                                                                             √       dx = (x ± 2a) x ± a + C                  (23)
                                                                                                 x±a     3
                          (x + a)n+1 ((n + 1)x − a)
           x(x + a)n dx =                           +c              (7)
                               (n + 1)(n + 2)
                                                                                               x
                             1                                                                    dx = −         x(a − x)
                                 dx = tan−1 x + c                   (8)                       a−x
                          1 + x2
                                                                                                                       x(a − x)
                                                                                                          − a tan−1             +C            (24)
                         1        1     x                                                                              x−a
                              dx = tan−1 + c                        (9)
                      a2 + x2     a     a
                         x        1                                                            x
                              dx = ln |a2 + x2 | + c               (10)                           dx =         x(a + x)
                    a2   +x 2     2                                                           a+x
                                                                                                                 √     √
                         x2                   x                                                           − a ln x + x + a + C                (25)
                              dx = x − a tan−1 + c                 (11)
                    a2   +x 2                 a
                   x3       1    1                                                       √
                        dx = x2 − a2 ln |a2 + x2 | + c             (12)
              a2   + x2     2    2                                                      x ax + bdx =
                                                                                          2                         √
           1                2             2ax + b                                           2
                                                                                              (−2b2 + abx + 3a2 x2 ) ax + b + C               (26)
                   dx = √          tan−1 √          + C (13)                             15a
     ax 2 + bx + c               2
                          4ac − b          4ac − b2

                   1              1    a+x                                                                  1
                            dx =    ln     , a=b                   (14)             x(ax + b)dx =               (2ax + b) ax(ax + b)
             (x + a)(x + b)      b−a b+x                                                                 4a3/2
                                                                                                               √
                   x            a                                                                      −b2 ln a x + a(ax + b) + C             (27)
                        2
                          dx =     + ln |a + x| + C                (15)
                (x + a)        a+x

           x             1                                                                               b     b2  x
                   dx =    ln |ax2 + bx + c|                                      x3 (ax + b)dx =          − 2 +       x3 (ax + b)
    ax2   + bx + c      2a                                                                             12a 8a x 3
                             b               2ax + b                                                   b3       √
                      − √            tan−1 √           + C (16)                                      + 5/2 ln a x + a(ax + b) + C (28)
                        a 4ac − b  2          4ac − b2                                                8a
   ∗ c 2007. From http://integral-table.com, last revised July 5, 2009. This material is provided as is without warranty or representation about

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                                                                          1
                                                                                      Integrals with Logarithms
                               1
              x2 ± a2 dx =       x x2 ± a2
                               2                                                           ln axdx = x ln ax − x + C                (41)
                               1
                              ± a2 ln x + x2 ± a2 + C          (29)
                               2                                                           ln ax     1       2
                                                                                                 dx = (ln ax) + C                   (42)
                                                                                             x       2
                               1
               a2 − x2 dx =      x a2 − x2                                                         b
                               2                                          ln(ax + b)dx =      x+       ln(ax + b) − x + C, a = 0 (43)
                               1             x                                                     a
                              + a2 tan−1 √         +C          (30)
                               2           a2 − x2


                                    1 2          3/2                            ln a2 x2 ± b2 dx = x ln a2 x2 ± b2
               x    x2 ± a2 dx =      x ± a2           +C      (31)
                                    3                                                                      2b       ax
                                                                                                       +      tan−1    − 2x + C     (44)
                   1                                                                                       a         b
              √          dx = ln x +         x2 ± a2 + C       (32)
                x 2 ± a2


                             1               x                                  ln a2 − b2 x2 dx = x ln ar − b2 x2
                       √           dx = sin−1 + C              (33)
                           a2 − x2           a
                                                                                                           2a       bx
                                                                                                       +      tan−1    − 2x + C     (45)
                          x                                                                                 b        a
                    √           dx =      x2 ± a2 + C          (34)
                        x2 ± a2
                         x                                                                             1                    2ax + b
                   √           dx = −      a2 − x2 + C         (35)        ln ax2 + bx + c dx =             4ac − b2 tan−1 √
                       a2 − x2                                                                         a                     4ac − b2
                                                                                           b
                                                                             − 2x +          + x ln ax2 + bx + c + C                (46)
                                                                                          2a
              x2         1
          √          dx = x x2 ± a2
            x 2 ± a2     2
                         1 2                                                                    bx 1 2
                           a ln x + x2 ± a2 + C                (36)         x ln(ax + b)dx =      − x
                         2                                                                      2a 4
                                                                                                1     b2
                                                                                              +   x2 − 2         ln(ax + b) + C     (47)
                                                                                                2     a
                                b + 2ax
    ax2 + bx + cdx =                       ax2 + bx + c
                                   4a
                                                                                                1
        4ac − b2                                                          x ln a2 − b2 x2 dx = − x2 +
+                ln 2ax + b + 2           a(ax2 + bx+ c) + C   (37)                             2
         8a3/2
                                                                                            1       a2
                                                                                               x2 − 2           ln a2 − b2 x2 + C (48)
                                                                                            2       b
                     1      √
x    ax2 + bx + c =   5/2
                          2 a ax2 + bx + c                                          Integrals with Exponentials
                   48a
− 3b2 + 2abx + 8a(c + ax2 )
                            √                                                                              1 ax
                                                                                              eax dx =       e +C                   (49)
+3(b3 − 4abc) ln b + 2ax + 2 a ax2 + bx + x                    (38)                                        a

                                                                                                       √
                                                                              √     ax    1 √ ax      i π      √
                    1                                                             xe dx =    xe + 3/2 erf i ax + C,
          √                  dx =                                                         a          2a
                                                                                                             x
              ax2   + bx + c                                                                            2         2

           1                                                                            where erf(x) = √       e−t dtet             (50)
          √ ln 2ax + b + 2           a(ax2 + bx + c) + C       (39)                                      π 0
            a
                                                                                            xex dx = (x − 1)ex + C                  (51)
              x             1
    √                  dx =         ax2 + bx + c                                                       x   1
        ax2   + bx + c      a                                                            xeax dx =       −        eax + C           (52)
       b                                                                                               a a2
    + 3/2 ln 2ax + b + 2               a(ax2 + bx + c) + C     (40)
     2a
                                                                                         x2 ex dx = x2 − 2x + 2 ex + C              (53)


                                                                      2
                           x2  2x   2                                                                 1
       x2 eax dx =            − 2 + 3      eax + C       (54)                      sin2 x cos xdx =     sin3 x + C           (68)
                           a   a   a                                                                  3

       x3 ex dx = x3 − 3x2 + 6x − 6 ex + C               (55)                                  cos[(2a − b)x] cos bx
                                                                          cos2 ax sin bxdx =                  −
                                                                                                 4(2a − b)       2b
                                                                                               cos[(2a + b)x]
                     (−1)n                                                                   −                +C             (69)
      xn eax dx =          Γ[1 + n, −ax],                                                        4(2a + b)
                      an+1
                                       ∞
                                                                                                       1
                   where Γ(a, x) =         ta−1 e−t dt   (56)                cos2 ax sin axdx = −        cos3 ax + C         (70)
                                      x                                                               3a
                        √
                2      i π      √
             eax dx = − √ erf ix a                       (57)                              x sin 2ax sin[2(a − b)x]
                       2 a                                          sin2 ax cos2 bxdx =       −       −
                                                                                           4      8a       16(a − b)
Integrals with Trigonometric Functions                                                     sin 2bx sin[2(a + b)x]
                                                                                         +        −               +C         (71)
                                                                                              8b     16(a + b)
                         1
             sin axdx = − cos ax + C                     (58)                                         x sin 4ax
                         a                                                  sin2 ax cos2 axdx =         −       +C           (72)
                                                                                                      8   32a
                             x sin 2ax
            sin2 axdx =        −       +C                (59)                                  1
                                                                                   tan axdx = − ln cos ax + C                (73)
                             2    4a                                                           a
                                                                                                       1
        n                                                                      tan2 axdx = −x +          tan ax + C          (74)
      sin axdx =                                                                                       a
       1                    1 1−n 3
      − cos ax      2 F1      ,   , , cos2 ax + C        (60)                               tann+1 ax
       a                    2   2  2                                        tann axdx =                 ×
                                                                                              a(1 + n)
                           3 cos ax cos 3ax                                          n+1      n+3
       sin3 axdx = −               +        +C           (61)               2 F1         , 1,       , − tan2 ax + C          (75)
                              4a      12a                                             2         2
                               1                                                          1              1
              cos axdx =         sin ax + C              (62)             tan3 axdx =       ln cos ax +    sec2 ax + C       (76)
                               a                                                          a             2a
                             x sin 2ax
            cos2 axdx =        +       +C                (63)
                             2    4a
                                                                               sec xdx = ln | sec x + tan x| + C
                                                                                                            x
                        1                                                                = 2 tanh−1 tan       +C             (77)
cosp axdx = −                cos1+p ax×                                                                     2
                    a(1 + p)
                    1+p 1 3+p                                                                      1
            2 F1          , ,       , cos2 ax + C        (64)                        sec2 axdx =     tan ax + C              (78)
                      2    2    2                                                                  a

                           3 sin ax sin 3ax
        cos3 axdx =                +        +C           (65)                      1              1
                              4a      12a                           sec3 x dx =      sec x tan x + ln | sec x + tan x| + C   (79)
                                                                                   2              2

                              cos[(a − b)x]
       cos ax sin bxdx =                    −                                        sec x tan xdx = sec x + C               (80)
                                 2(a − b)
                           cos[(a + b)x]
                                         + C, a = b      (66)                                         1
                             2(a + b)                                              sec2 x tan xdx =     sec2 x + C           (81)
                                                                                                      2
                                                                                                 1
  2                 sin[(2a − b)x]                                          secn x tan xdx =       secn x + C, n = 0         (82)
sin ax cos bxdx = −                                                                              n
                       4(2a − b)
                  sin bx sin[(2a + b)x]
                +        −              +C               (67)                            x
                    2b       4(2a + b)                              csc xdx = ln tan       + C = ln | csc x − cot x| + C     (83)
                                                                                         2

                                                                3
                                                                      Products of Trigonometric Functions and
                     2        1                                                    Exponentials
                  csc axdx = − cot ax + C                (84)
                              a
                                                                                             1 x
                                                                              ex sin xdx =     e (sin x − cos x) + C         (99)
               1             1                                                               2
   csc3 xdx = − cot x csc x + ln | csc x − cot x| + C (85)
               2             2
                                                                                        1
                                                                    ebx sin axdx =           ebx (b sin ax − a cos ax) + C (100)
                                 1                                                   a2 + b2
           cscn x cot xdx = −      cscn x + C, n = 0     (86)
                                 n                                                           1 x
                                                                              ex cos xdx =     e (sin x + cos x) + C        (101)
                                                                                             2
                 sec x csc xdx = ln | tan x| + C         (87)
                                                                                        1
                                                                    ebx cos axdx =           ebx (a sin ax + b cos ax) + C (102)
Products of Trigonometric Functions and Monomials                                    a2 + b2

                                                                                     1 x
                                                                     xex sin xdx =     e (cos x − x cos x + x sin x) + C    (103)
                x cos xdx = cos x + x sin x + C          (88)                        2

                          1          x                                               1 x
           x cos axdx =      cos ax + sin ax + C         (89)        xex cos xdx =     e (x cos x − sin x + x sin x) + C    (104)
                          a2         a                                               2
                                                                           Integrals of Hyperbolic Functions
          2                           2
         x cos xdx = 2x cos x + x − 2 sin x + C          (90)
                                                                                                 1
                                                                                  cosh axdx =      sinh ax + C              (105)
                                                                                                 a
                                    2 2
                     2x cos ax a x − 2
     x2 cos axdx =            +        sin ax + C        (91)
                        a2        a3
                                                                     eax cosh bxdx =
                                                                          ax
                                                                          e      [a cosh bx − b sinh bx] + C      a=b
                        1                                                 2
                                                                           a − b2
           xn cosxdx = − (i)n+1 [Γ(n + 1, −ix)                              2ax                                             (106)
                        2                                                e       x
                                                                               + +C                               a=b
                     +(−1)n Γ(n + 1, ix)] + C            (92)               4a     2
                                                                                                 1
                                                                                  sinh axdx =      cosh ax + C              (107)
                                                                                                 a
                       1
       xn cosaxdx =      (ia)1−n [(−1)n Γ(n + 1, −iax)
                       2                                            eax sinh bxdx =
                     −Γ(n + 1, ixa)] + C                 (93)            ax
                                                                         e
                                                                         2       [−b cosh bx + a sinh bx] + C     a=b
                                                                           a − b2                                           (108)
                                                                            2ax
                                                                        e        x
                x sin xdx = −x cos x + sin x + C         (94)                  − +C                               a=b
                                                                            4a    2

                               x cos ax sin ax
              x sin axdx = −           +       +C        (95)       eax tanh bxdx =
                                  a       a2
                                                                     (a+2b)x
                                                                    e                  a             a     2bx
                                                                     (a + 2b) 2 F1 1 + 2b , 1, 2 + 2b , −e
                                                                    
         x2 sin xdx = 2 − x2 cos x + 2x sin x + C
                                                                    
                                                         (96)       
                                                                    
                                                                               1         a
                                                                             − eax 2 F1      , 1, 1E, −e2bx + C     a = b (109)
                                                                              a −1 ax 2b
                                                                     eax − 2 tan [e ]
                                                                    
                                                                    
                     2 − a2 x2                                                           +C                         a=b
                                                                    
                                        2x sin ax                   
     x2 sin axdx =        3
                               cos ax +           +C     (97)                  a
                        a                  a2
                                                                                                1
                                                                                 tanh bxdx =      ln cosh ax + C            (110)
                                                                                                a
                         1
           xn sin xdx = − (i)n [Γ(n + 1, −ix)                                                     1
                         2                                                cos ax cosh bxdx =            [a sin ax cosh bx
                      −(−1)n Γ(n + 1, −ix)] + C          (98)                                  a2 + b2
                                                                                             +b cos ax sinh bx] + C         (111)


                                                                4
                        1                                                            1
 cos ax sinh bxdx =          [b cos ax cosh bx+               sinh ax cosh axdx =      [−2ax + sinh 2ax] + C    (115)
                      a2+ b2                                                        4a
                 a sin ax sinh bx] + C          (112)

                                                                                         1
                       1                                         sinh ax cosh bxdx =          [b cosh bx sinh ax
sin ax cosh bxdx =          [−a cos ax cosh bx+                                       b2 − a2
                    a2 + b2
                                                                                    −a cosh ax sinh bx] + C      (116)
                b sin ax sinh bx] + C           (113)

                       1
 sin ax sinh bxdx =          [b cosh bx sin ax−
                    a2 + b2
                 a cos ax sinh bx] + C            (114)




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