Table of Integrals
∗
Basic Forms
Z Integrals of Rational Functions
Z 1 Integrals with Roots
Z √
Integrals with Logarithms
Z
Integrals with Exponentials
Z
∗« 2014. From http://integral-table.com, last revised June 14, 2014. This material is provided as is without warranty or representation about the accuracy, correctness or
Integrals with Trigonometric Functions
Z
sinaxdx=−1
acosax (63)
Z
sin2axdx= x 2−
sin 2ax
4a (64)
Z
sinnaxdx=
−1
acosax 2F1
1 2,
1−n
2 , 3 2,cos
2 ax
(65)
Z
sin3axdx=−3 cos4 ax
a +
cos 3ax
12a (66)
Z
cosaxdx=1
asinax (67)
Z
cos2axdx=x 2+
sin 2ax
4a (68)
Z
cospaxdx=− 1
a(1 +p)cos
1+pax
×
2F1
1 +p
2 , 1 2,
3 +p
2 ,cos
2ax
(69)
Z
cos3axdx= 3 sinax 4a +
sin 3ax
12a (70)
Z
cosaxsinbxdx= cos[(a−b)x] 2(a−b) −
cos[(a+b)x] 2(a+b) , a6=b
(71)
Z
sin2axcosbxdx=−sin[(24(2a−b)x]
a−b) +sinbx
2b −
sin[(2a+b)x]
4(2a+b) (72) Z
sin2xcosxdx=1 3sin
3
x (73)
Z
cos2axsinbxdx=cos[(2a−b)x] 4(2a−b) −
cosbx
2b
−cos[(24(2a+b)x]
a+b) (74) Z
cos2axsinaxdx=−31acos3ax (75)
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) (76) Z
sin2axcos2axdx=x 8 −
sin 4ax
32a (77)
Z
tanaxdx=−1
aln cosax (78)
Z
tan2axdx=−x+1
atanax (79)
Z
tannaxdx=tan
n+1ax a(1 +n) ×
2F1
n+ 1 2 ,1,
n+ 3 2 ,−tan
2ax
(80)
Z
tan3axdx=1
aln cosax+
1 2asec
2
ax (81)
Z
secxdx= ln|secx+ tanx|= 2 tanh−1
tanx 2
(82)
Z
sec2axdx= 1
atanax (83)
Z
sec3xdx =1
2secxtanx+ 1
2ln|secx+ tanx| (84) Z
secxtanxdx= secx (85) Z
sec2xtanxdx=1 2sec
2
x (86)
Z
secnxtanxdx= 1
nsec n
x, n6= 0 (87)
Z
cscxdx= ln tan
x
2
= ln|cscx−cotx|+C (88) Z
csc2axdx=−1
acotax (89)
Z
csc3xdx=−12cotxcscx+1
2ln|cscx−cotx| (90) Z
cscnxcotxdx=−1
ncsc n
x, n6= 0 (91) Z
secxcscxdx= ln|tanx| (92)
Products of Trigonometric Functions and Monomials
Z
xcosxdx= cosx+xsinx (93) Z
xcosaxdx= 1
a2cosax+ x
asinax (94)
Z
x2cosxdx= 2xcosx+ x2−2
sinx (95)
Z
x2cosaxdx= 2xcosax
a2 +
a2x2−2
a3 sinax (96)
Z
xncosxdx=−12(i)n+1[Γ(n+ 1,−ix)
+(−1)nΓ(n+ 1, ix)] (97)
Z
xncosaxdx= 1 2(ia)
1−n
[(−1)nΓ(n+ 1,−iax) −Γ(n+ 1, ixa)] (98)
Z
xsinxdx=−xcosx+ sinx (99) Z
xsinaxdx=−xcosax
a +
sinax
a2 (100)
Z
x2sinxdx= 2−x2
cosx+ 2xsinx (101)
Z
x2sinaxdx= 2−a
2x2
a3 cosax+
2xsinax
a2 (102)
Z
xnsinxdx=−12(i)n[Γ(n+ 1,−ix)−(−1)nΓ(n+ 1,−ix)] (103)
Products of Trigonometric Functions and Exponentials
Z
exsinxdx=1 2e
x(sinx
−cosx) (104)
Z
ebxsinaxdx= 1
a2+b2e bx(
bsinax−acosax) (105)
Z
excosxdx=1 2e
x(sinx+ cosx) (106)
Z
ebxcosaxdx= 1
a2+b2e bx(
asinax+bcosax) (107)
Z
xexsinxdx=1 2e
x
(cosx−xcosx+xsinx) (108)
Z
xexcosxdx= 1 2e
x(
xcosx−sinx+xsinx) (109)
Integrals of Hyperbolic Functions
Z
coshaxdx=1
asinhax (110)
Z
eaxcoshbxdx=
eax
a2−b2[acoshbx−bsinhbx] a6=b e2ax
4a + x
2 a=b
(111)
Z
sinhaxdx= 1
acoshax (112)
Z
eaxsinhbxdx=
eax
a2−b2[−bcoshbx+asinhbx] a6=b e2ax
4a − x
2 a=b
(113)
Z
eaxtanhbxdx=
e(a+2b)x
(a+ 2b)2F1 h
1 + a 2b,1,2 +
a
2b,−e 2bxi
−1
ae ax
2F1ha
2b,1,1E,−e 2bxi
a6=b eax−2 tan−1
[eax]
a a=b
(114)
Z
tanhax dx=1
aln coshax (115)
Z
cosaxcoshbxdx= 1
a2+b2[asinaxcoshbx
+bcosaxsinhbx] (116)
Z
cosaxsinhbxdx= 1
a2+b2[bcosaxcoshbx+ asinaxsinhbx] (117)
Z
sinaxcoshbxdx= 1
a2+b2[−acosaxcoshbx+ bsinaxsinhbx] (118)
Z
sinaxsinhbxdx= 1
a2+b2[bcoshbxsinax− acosaxsinhbx] (119)
Z
sinhaxcoshaxdx= 1
4a[−2ax+ sinh 2ax] (120)
Z
sinhaxcoshbxdx= 1
b2−a2[bcoshbxsinhax
