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# Integration Table

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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)

5

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