Integral table

integrals and derivatives

1. Table of Integrals
*
Basic Forms
Z
x
n
dx=
1
n+ 1
x
n+1
(1)
Z
1
x
dx= lnjxj (2)
Z
udv=uv
Z
vdu (3)
Z
1
ax+b
dx=
1
a
lnjax+bj (4)
Integrals of Rational Functions
Z
1
(x+a)
2
dx=
1
x+a
(5)
Z
(x+a)
n
dx=
(x+a)
n+1
n+ 1
+c; n6=1 (6)
Z
x(x+a)
n
dx=
(x+a)
n+1
((n+ 1)xa)
(n+ 1)(n+ 2)
(7)
Z
1
1 +x
2
dx= tan
1
x (8)
Z
1
a
2
+x
2
dx=
1
a
tan
1x
a
(9)
Z
x
a
2
+x
2
dx=
1
2
lnja
2
+x
2
j (10)
Z
x
2
a
2
+x
2
dx=xatan
1x
a
(11)
Z
x
3
a
2
+x
2
dx=
1
2
x
2

1
2
a
2
lnja
2
+x
2
j (12)
Z
1
ax
2
+bx+c
dx=
2
p
4acb
2
tan
12ax+b
p
4acb
2
(13)
Z
1
(x+a)(x+b)
dx=
1
ba
ln
a+x
b+x
; a6=b (14)
Z
x
(x+a)
2
dx=
a
a+x
+ lnja+xj (15)
Z
x
ax
2
+bx+c
dx=
1
2a
lnjax
2
+bx+cj

b
a
p
4acb
2
tan
12ax+b
p
4acb
2
(16)
Integrals with Roots
Z
p
xadx=
2
3
(xa)
3=2
(17)
Z
1
p
xa
dx= 2
p
xa (18)
Z
1
p
ax
dx=2
p
ax (19)
Z
x
p
xadx=
2
3
a(xa)
3=2
+
2
5
(xa)
5=2
(20)
Z
p
ax+bdx=

2b
3a
+
2x
3

p
ax+b (21)
Z
(ax+b)
3=2
dx=
2
5a
(ax+b)
5=2
(22)
Z
x
p
xa
dx=
2
3
(x2a)
p
xa (23)
Zr
x
ax
dx=
p
x(ax)atan
1
p
x(ax)
xa
(24)
Zr
x
a+x
dx=
p
x(a+x)aln
p
x+
p
x+a

(25)
Z
x
p
ax+bdx=
2
15a
2
(2b
2
+abx+ 3a
2
x
2
)
p
ax+b(26)
Z
p
x(ax+b)dx=
1
4a
3=2
h
(2ax+b)
p
ax(ax+b)
b
2
ln


a
p
x+
p
a(ax+b)



i
(27)
Z
p
x
3
(ax+b)dx=

b
12a

b
2
8a
2
x
+
x
3

p
x
3
(ax+b)
+
b
3
8a
5=2
ln


a
p
x+
p
a(ax+b)


(28)
Z
p
x
2
a
2
dx=
1
2
x
p
x
2
a
2

1
2
a
2
ln


x+
p
x
2
a
2



(29)
Z
p
a
2
x
2
dx=
1
2
x
p
a
2
x
2
+
1
2
a
2
tan
1 x
p
a
2
x
2
(30)
Z
x
p
x
2
a
2
dx=
1
3

x
2
a
2

3=2
(31)
Z
1
p
x
2
a
2
dx= ln


x+
p
x
2
a
2


(32)
Z
1
p
a
2
x
2
dx= sin
1x
a
(33)
Z
x
p
x
2
a
2
dx=
p
x
2
a
2
(34)
Z
x
p
a
2
x
2
dx=
p
a
2
x
2
(35)
Z
x
2
p
x
2
a
2
dx=
1
2
x
p
x
2
a
2

1
2
a
2
ln


x+
p
x
2
a
2



(36)
Z
p
ax
2
+bx+cdx=
b+ 2ax
4a
p
ax
2
+bx+c
+
4acb
2
8a
3=2
ln


2ax+b+ 2
p
a(ax
2
+bx
+
c)


(37)
Z
x
p
ax
2
+bx+c=
1
48a
5=2

2
p
a
p
ax
2
+bx+c


3b
2
+ 2abx+ 8a(c+ax
2
)

+3(b
3
4abc) ln


b+ 2ax+ 2
p
a
p
ax
2
+bx+x




(38)
Z
1
p
ax
2
+bx+c
dx=
1
p
a
ln


2ax+b+ 2
p
a(ax
2
+bx+c)



(39)
Z
x
p
ax
2
+bx+c
dx=
1
a
p
ax
2
+bx+c

b
2a
3=2
ln


2ax+b+ 2
p
a(ax
2
+bx+c)


(40)
Z
dx
(a
2
+x
2
)
3=2
=
x
a
2
p
a
2
+x
2
(41)
Integrals with Logarithms
Z
lnaxdx=xlnaxx (42)
Z
lnax
x
dx=
1
2
(lnax)
2
(43)
Z
ln(ax+b)dx=

x+
b
a

ln(ax+b)x; a6= 0 (44)
Z
ln(x
2
+a
2
) dx =xln(x
2
+a
2
) + 2atan
1x
a
2x(45)
Z
ln(x
2
a
2
) dx =xln(x
2
a
2
) +aln
x+a
xa
2x(46)
Z
ln

ax
2
+bx+c

dx=
1
a
p
4acb
2
tan
12ax+b
p
4acb
2
2x+

b
2a
+x

ln

ax
2
+bx+c

(47)
Z
xln(ax+b)dx=
bx
2a

1
4
x
2
+
1
2

x
2

b
2
a
2

ln(ax+b) (48)
Z
xln

a
2
b
2
x
2

dx=
1
2
x
2
+
1
2

x
2

a
2
b
2

ln

a
2
b
2
x
2

(49)
Integrals with Exponentials
Z
e
ax
dx=
1
a
e
ax
(50)
Z
p
xe
ax
dx=
1
a
p
xe
ax
+
i
p

2a
3=2
erf

i
p
ax

;
where erf(x) =
2
p

Z
x
0
e
t
2
dt(51)
Z
xe
x
dx= (x1)e
x
(52)
Z
xe
ax
dx=

x
a

1
a
2

e
ax
(53)
Z
x
2
e
x
dx=

x
2
2x+ 2

e
x
(54)
Z
x
2
e
ax
dx=

x
2
a

2x
a
2
+
2
a
3

e
ax
(55)
Z
x
3
e
x
dx=

x
3
3x
2
+ 6x6

e
x
(56)
Z
x
n
e
ax
dx=
x
n
e
ax
a

n
a
Z
x
n1
e
ax
dx (57)
Z
x
n
e
ax
dx=
(1)
n
a
n+1
[1 +n;ax];
where (a; x) =
Z
1
x
t
a1
e
t
dt
(58)
Z
e
ax
2
dx=
i
p

2
p
a
erf

ix
p
a

(59)
Z
e
ax
2
dx=
p

2
p
a
erf

x
p
a

(60)
Z
xe
ax
2
dx =
1
2a
e
ax
2
(61)
Z
x
2
e
ax
2
dx =
1
4
r

a
3
erf(x
p
a)
x
2a
e
ax
2
(62)
*
«2012. Fromhttp://integral-table.com, last revised September 8, 2012. This material 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 Commons 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/or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco,
California, 94105, USA.
1

2. Integrals with Trigonometric Functions
Z
sinaxdx=
1
a
cosax (63)
Z
sin
2
axdx=
x
2

sin 2ax
4a
(64)
Z
sin
n
axdx=

1
a
cosax2F1

1
2
;
1n
2
;
3
2
;cos
2
ax

(65)
Z
sin
3
axdx=
3 cosax
4a
+
cos 3ax
12a
(66)
Z
cosaxdx=
1
a
sinax (67)
Z
cos
2
axdx=
x
2
+
sin 2ax
4a
(68)
Z
cos
p
axdx=
1
a(1 +p)
cos
1+p
ax
2F1

1 +p
2
;
1
2
;
3 +p
2
;cos
2
ax

(69)
Z
cos
3
axdx=
3 sinax
4a
+
sin 3ax
12a
(70)
Z
cosaxsinbxdx=
cos[(ab)x]
2(ab)

cos[(a+b)x]
2(a+b)
; a6=b
(71)
Z
sin
2
axcosbxdx=
sin[(2ab)x]
4(2ab)
+
sinbx
2b

sin[(2a+b)x]
4(2a+b)
(72)
Z
sin
2
xcosxdx=
1
3
sin
3
x (73)
Z
cos
2
axsinbxdx=
cos[(2ab)x]
4(2ab)

cosbx
2b

cos[(2a+b)x]
4(2a+b)
(74)
Z
cos
2
axsinaxdx=
1
3a
cos
3
ax (75)
Z
sin
2
axcos
2
bxdx=
x
4

sin 2ax
8a

sin[2(ab)x]
16(ab)
+
sin 2bx
8b

sin[2(a+b)x]
16(a+b)
(76)
Z
sin
2
axcos
2
axdx=
x
8

sin 4ax
32a
(77)
Z
tanaxdx=
1
a
ln cosax (78)
Z
tan
2
axdx=x+
1
a
tanax (79)
Z
tan
n
axdx=
tan
n+1
ax
a(1 +n)

2F1

n+ 1
2
;1;
n+ 3
2
;tan
2
ax

(80)
Z
tan
3
axdx=
1
a
ln cosax+
1
2a
sec
2
ax (81)
Z
secxdx= lnjsecx+ tanxj= 2 tanh
1

tan
x
2

(82)
Z
sec
2
axdx=
1
a
tanax (83)
Z
sec
3
xdx =
1
2
secxtanx+
1
2
lnjsecx+ tanxj(84)
Z
secxtanxdx= secx (85)
Z
sec
2
xtanxdx=
1
2
sec
2
x (86)
Z
sec
n
xtanxdx=
1
n
sec
n
x; n6= 0 (87)
Z
cscxdx= ln


tan
x
2


= lnjcscxcotxj+C (88)
Z
csc
2
axdx=
1
a
cotax (89)
Z
csc
3
xdx=
1
2
cotxcscx+
1
2
lnjcscxcotxj(90)
Z
csc
n
xcotxdx=
1
n
csc
n
x; n6= 0 (91)
Z
secxcscxdx= lnjtanxj (92)
Products of Trigonometric Functions and
Monomials
Z
xcosxdx= cosx+xsinx (93)
Z
xcosaxdx=
1
a
2
cosax+
x
a
sinax (94)
Z
x
2
cosxdx= 2xcosx+

x
2
2

sinx (95)
Z
x
2
cosaxdx=
2xcosax
a
2
+
a
2
x
2
2
a
3
sinax (96)
Z
x
n
cosxdx=
1
2
(i)
n+1
[(n+ 1;ix)
+(1)
n
(n+ 1; ix)] (97)
Z
x
n
cosaxdx=
1
2
(ia)
1n
[(1)
n
(n+ 1;iax)
(n+ 1; ixa)] (98)
Z
xsinxdx=xcosx+ sinx (99)
Z
xsinaxdx=
xcosax
a
+
sinax
a
2
(100)
Z
x
2
sinxdx=

2x
2

cosx+ 2xsinx (101)
Z
x
2
sinaxdx=
2a
2
x
2
a
3
cosax+
2xsinax
a
2
(102)
Z
x
n
sinxdx=
1
2
(i)
n
[(n+ 1;ix)(1)
n
(n+ 1;ix)]
(103)
Products of Trigonometric Functions and
Exponentials
Z
e
x
sinxdx=
1
2
e
x
(sinxcosx) (104)
Z
e
bx
sinaxdx=
1
a
2
+b
2
e
bx
(bsinaxacosax) (105)
Z
e
x
cosxdx=
1
2
e
x
(sinx+ cosx) (106)
Z
e
bx
cosaxdx=
1
a
2
+b
2
e
bx
(asinax+bcosax) (107)
Z
xe
x
sinxdx=
1
2
e
x
(cosxxcosx+xsinx) (108)
Z
xe
x
cosxdx=
1
2
e
x
(xcosxsinx+xsinx) (109)
Integrals of Hyperbolic Functions
Z
coshaxdx=
1
a
sinhax (110)
Z
e
ax
coshbxdx=
8
>
<
>
:
e
ax
a
2
b
2
[acoshbxbsinhbx]a6=b
e
2ax
4a
+
x
2
a=b
(111)
Z
sinhaxdx=
1
a
coshax (112)
Z
e
ax
sinhbxdx=
8
>
<
>
:
e
ax
a
2
b
2
[bcoshbx+asinhbx]a6=b
e
2ax
4a

x
2
a=b
(113)
Z
e
ax
tanhbxdx=
8
>
>
>
>
>
<
>
>
>
>
>
:
e
(a+2b)x
(a+ 2b)
2F1
h
1 +
a
2b
;1;2 +
a
2b
;e
2bx
i

1
a
e
ax
2F1
h
a
2b
;1;1E;e
2bx
i
a6=b
e
ax
2 tan
1
[e
ax
]
a
a=b
(114)
Z
tanhax dx=
1
a
ln coshax (115)
Z
cosaxcoshbxdx=
1
a
2
+b
2
[asinaxcoshbx
+bcosaxsinhbx] (116)
Z
cosaxsinhbxdx=
1
a
2
+b
2
[bcosaxcoshbx+
asinaxsinhbx] (117)
Z
sinaxcoshbxdx=
1
a
2
+b
2
[acosaxcoshbx+
bsinaxsinhbx] (118)
Z
sinaxsinhbxdx=
1
a
2
+b
2
[bcoshbxsinax
acosaxsinhbx] (119)
Z
sinhaxcoshaxdx=
1
4a
[2ax+ sinh 2ax] (120)
Z
sinhaxcoshbxdx=
1
b
2
a
2
[bcoshbxsinhax
acoshaxsinhbx] (121)
2