Table of Integrals
Basic Forms
(1)
Z
xndx= 1
n+ 1x
n+1, n
6
=−1
(2)
Z
1
xdx= ln|x|
(3)
Z
udv=uv −
Z
vdu
(4)
Z
1
ax+bdx=
1
aln|ax+b|
Integrals of Rational Functions
(5)
Z
1
(x+a)2dx=−
1
x+a
(6)
Z
(x+a)ndx= (x+a)
n+1
n+ 1 +c, n6=−1
(7)
Z
x(x+a)ndx= (x+a)
n+1((n+ 1)x−a)
(n+ 1)(n+ 2)
(8)
Z
1
1 +x2dx= tan
−1
x
(9)
Z 1
a2+x2dx=
1
atan
−1 x
(10)
Z
x
a2+x2dx=
1 2ln|a
2+x2
|
(11)
Z
x2
a2 +x2dx=x−atan
−1 x
a
(12)
Z
x3
a2+x2dx=
1 2x
2
− 12a2ln|a2+x2|
(13)
Z 1
ax2 +bx+cdx=
2
√
4ac−b2 tan
−1 2ax+b
√
4ac−b2
(14)
Z 1
(x+a)(x+b)dx= 1
b−aln a+x
b+x, a6=b
(15)
Z x
(x+a)2dx=
a
a+x + ln|a+x|
(16)
Z x
ax2+bx+cdx=
1 2aln|ax
2+bx+c
|− b
a√4ac−b2 tan
−1 2ax+b
√
4ac−b2
Integrals with Roots
(17)
Z √
x−a dx= 2
3(x−a)
3/2
(18)
Z
1
√
x±a dx= 2
√
x±a
(19)
Z 1
√
a−x dx=−2
√
(30)
Z √
a2−x2 dx= 1
2x
√
a2−x2+ 1
2a
2tan−1 x
√
a2−x2
(31)
Z
x√x2±a2 dx= 1
3 x
2±a23/2
(32)
Z
1
√
x2 ±a2 dx= ln
x+
√
x2±a2
(33)
Z 1
√
a2−x2 dx= sin
−1 x
a
(34)
Z x
√
x2±a2 dx=
√
x2±a2
(35)
Z x
√
a2−x2 dx=−
√
a2−x2
(36)
Z x2
√
x2±a2 dx=
1 2x
√
x2±a2∓1
2a
2ln x+
√
x2±a2
(37)
Z √
ax2+bx+c dx= b+ 2ax
4a
√
ax2+bx+c+4ac−b 2
8a3/2 ln
2ax+b+ 2 p
a(ax2+bx+c)
Z
x√ax2+bx+c dx = 1
48a5/2
2√a√ax2+bx+c −3b2+ 2abx+ 8a(c+ax2)
+3(b3−4abc) ln
b+ 2ax+ 2 √
a√ax2+bx+x
(39)
Integrals with Logarithms
(49)
Z
ln(x2+a2)dx=xln(x2+a2) + 2atan−1 x
a −2x
(50)
Z
ln(x2−a2)dx=xln(x2 −a2) +alnx+a
x−a −2x
(51)
Z
ln ax2+bx+c dx= 1
a
√
4ac−b2tan−1 2ax+b
√
4ac−b2−2x+
b
2a +x
ln ax2+bx+c
(52)
Z
xln(ax+b) dx= bx 2a −
1 4x
2+ 1
2
x2− b
2
a2
ln(ax+b)
(53)
Z
xln a2−b2x2
dx =−1 2x
2 +1
2
x2− a
2
b2
ln a2−b2x2
(54)
Z
(lnx)2 dx= 2x−2xlnx+x(lnx)2
(55)
Z
(lnx)3 dx=−6x+x(lnx)3−3x(lnx)2+ 6xlnx
(56)
Z
x(lnx)2 dx= x
2
4 +
1 2x
2(lnx)2
− 12x2lnx
(57)
Z
x2(lnx)2 dx= 2x
3
27 + 1 3x
3(lnx)2
Integrals with Exponentials
(58)
Z
eax dx= 1
ae
ax
(59)
Z √
xeax dx= 1
a
√
xeax+ i
√π
2a3/2erf i
√
ax, where erf(x) = √2
π
Z x
0
e−t2
dt
(60)
Z
xex dx= (x−1)ex
(61)
Z
xeax dx=
x a −
1
a2
eax
(62)
Z
x2ex dx= x2−2x+ 2
ex
(63)
Z
x2eax dx=
x2 a −
2x a2 +
2
a3
eax
(64)
Z
x3ex dx= x3−3x2+ 6x−6
ex
(65)
Z
xneax dx= x
neax
a − n a
Z
xn−1
eaxdx
(66)
Z
xneax dx= (−1)
n
an+1 Γ[1 +n,−ax], where Γ(a, x) =
Z ∞
x
ta−1
e−t dt
(67)
Z
eax2 dx=−i
√
π
2√aerf ix
√
(68)
Z
e−ax2
dx=
√
π
2√aerf x
√
a
(69)
Z
xe−ax2
dx =− 1 2ae
−ax2
(70)
Z
x2e−ax2
dx= 1 4
r
π a3erf(x
√
a)− x 2ae
−ax2
Integrals with Trigonometric Functions
(71)
Z
sinax dx=−1
acosax
(72)
Z
sin2ax dx= x 2 −
sin 2ax
4a
(73)
Z
sin3ax dx=−3 cosax
4a +
cos 3ax
12a
(74)
Z
sinnax dx=−1
acosax 2F1
1 2,
1−n
2 , 3 2,cos
2ax
(75)
Z
cosax dx= 1
asinax
(76)
Z
cos2ax dx= x
2 +
sin 2ax
4a
(77)
Z
cos3axdx= 3 sinax
4a +
sin 3ax
(78)
Z
cospaxdx=− 1
a(1 +p)cos
1+pax
×2F1
1 +p
2 , 1 2,
3 +p
2 ,cos
2ax
(79)
Z
cosxsinx dx= 1 2sin
2x+c 1 =−
1 2cos
2x+c 2 =−
1
4cos 2x+c3
(80)
Z
cosaxsinbx dx = cos[(a−b)x] 2(a−b) −
cos[(a+b)x] 2(a+b) , a6=b
(81)
Z
sin2axcosbx dx=−sin[(2a−b)x] 4(2a−b) +
sinbx
2b −
sin[(2a+b)x] 4(2a+b)
(82)
Z
sin2xcosx dx= 1 3sin
3x
(83)
Z
cos2axsinbx dx= cos[(2a−b)x] 4(2a−b) −
cosbx
2b −
cos[(2a+b)x] 4(2a+b)
(84)
Z
cos2axsinax dx=− 1 3acos
3ax
(85)
Z
sin2axcos2bxdx= x 4−
sin 2ax
8a −
sin[2(a−b)x] 16(a−b) +
sin 2bx
8b −
sin[2(a+b)x] 16(a+b)
(86)
Z
sin2axcos2ax dx= x 8 −
sin 4ax
32a
(87)
Z
tanax dx =−1
(88)
Z
tan2ax dx=−x+1
atanax
(89)
Z
tannax dx= tan
n+1ax
a(1 +n) ×2F1
n+ 1 2 ,1,
n+ 3
2 ,−tan
2ax
(90)
Z
tan3axdx= 1
aln cosax+
1 2asec
2ax
(91)
Z
secx dx= ln|secx+ tanx|= 2 tanh−1 tanx
2
(92)
Z
sec2ax dx = 1
atanax
(93)
Z
sec3x dx= 1
2secxtanx+ 1
2ln|secx+ tanx|
(94)
Z
secxtanx dx= secx
(95)
Z
sec2xtanx dx= 1 2sec
2x
(96)
Z
secnxtanx dx= 1
n sec
nx, n
6
= 0
(97)
Z
cscx dx= ln tan
x
2
(98)
Z
csc2ax dx =−1
acotax
(99)
Z
csc3x dx=−1
2cotxcscx+ 1
2ln|cscx−cotx|
(100)
Z
cscnxcotx dx=−1
n csc
nx, n
6
= 0
(101)
Z
secxcscx dx= ln|tanx|
Products of Trigonometric Functions and Monomials
(102)
Z
xcosx dx= cosx+xsinx
(103)
Z
xcosax dx= 1
a2 cosax+
x a sinax
(104)
Z
x2cosx dx= 2xcosx+ x2 −2
sinx
(105)
Z
x2cosax dx= 2xcosax
a2 +
a2x2−2
a3 sinax
(106)
Z
xncosxdx=−1 2(i)
n+1[Γ(n+ 1,
−ix) + (−1)nΓ(n+ 1, ix)]
(107)
Z
xncosax dx = 1 2(ia)
1−n
(108)
Z
xsinx dx=−xcosx+ sinx
(109)
Z
xsinax dx=−xcosax
a +
sinax a2
(110)
Z
x2sinx dx= 2−x2cosx+ 2xsinx
(111)
Z
x2sinax dx= 2−a
2x2
a3 cosax+
2xsinax a2
(112)
Z
xnsinx dx=−1 2(i)
n[Γ(n+ 1,
−ix)−(−1)nΓ(n+ 1,−ix)]
(113)
Z
xcos2x dx= x
2
4 + 1
8cos 2x+ 1
4xsin 2x
(114)
Z
xsin2x dx= x2
4 −
1
8cos 2x− 1
4xsin 2x
(115)
Z
xtan2x dx=−x
2
2 + ln cosx+xtanx
(116)
Z
Products of Trigonometric Functions and Exponentials
(117)
Z
exsinx dx= 1 2e
x(sinx
−cosx)
(118)
Z
ebxsinax dx= 1
a2 +b2e
bx(bsinax
−acosax)
(119)
Z
excosx dx= 1 2e
x(sinx+ cosx)
(120)
Z
ebxcosax dx= 1
a2+b2e
bx(asinax+bcosax)
(121)
Z
xexsinx dx= 1 2e
x(cosx
−xcosx+xsinx)
(122)
Z
xexcosx dx= 1 2e
x(xcosx
−sinx+xsinx)
Integrals of Hyperbolic Functions
(123)
Z
coshax dx= 1
asinhax
(124)
Z
eaxcoshbx dx=
eax
a2−b2[acoshbx−bsinhbx] a 6=b
e2ax
4a + x
2 a =b
(125)
Z
sinhax dx= 1
(126)
Z
eaxsinhbx dx=
eax
a2−b2[−bcoshbx+asinhbx] a6=b
e2ax
4a − x
2 a=b
(127)
Z
tanhax dx= 1
aln coshax
(128)
Z
eaxtanhbx dx=
e(a+2b)x
(a+ 2b)2F1
h
1 + a
2b,1,2 + a
2b,−e
2bxi
−1aeax2F1h1, a
2b,1 + a
2b,−e
2bxi a
6
=b eax−2 tan−1
[eax]
a a=b
(129)
Z
cosaxcoshbx dx= 1
a2+b2 [asinaxcoshbx+bcosaxsinhbx]
(130)
Z
cosaxsinhbx dx = 1
a2+b2 [bcosaxcoshbx+asinaxsinhbx]
(131)
Z
sinaxcoshbx dx = 1
a2+b2 [−acosaxcoshbx+bsinaxsinhbx]
(132)
Z
sinaxsinhbx dx= 1
a2+b2 [bcoshbxsinax−acosaxsinhbx]
(133)
Z
sinhaxcoshaxdx= 1
4a[−2ax+ sinh 2ax]
(134)
Z
sinhaxcoshbx dx= 1
b2−a2 [bcoshbxsinhax−acoshaxsinhbx]
c
2012. Fromhttp://integral-table.com, last revised September 8, 2012. This mate-rial is provided as is without warranty or representation about the accuracy, correctness or suitability of this material for any purpose. This work is licensed under the Creative Com-mons Attribution-Noncommercial-No Derivative Works 3.0 United States License. To view a copy of this license, visithttp://creativecommons.org/licenses/by-nc-nd/3.0/us/
