INTEGRALS
ELEMENTARY FORMS
1.
Z
adx ¼ ax
2.
Z
a f ðxÞdx ¼ a
Z
f ðxÞdx
3.
Z
ðyÞdx ¼
Z
ðyÞ
y
0
dy, where y
0
¼
dy
dx
4.
Z
ðu þ vÞdx ¼
Z
udxþ
Z
vdx, where u and v are any functions of x
5.
Z
udv ¼ u
Z
dv
Z
vdu ¼ uv
Z
vdu
6.
Z
u
dv
dx
dx ¼ uv
Z
v
du
dx
dx
7.
Z
x
n
dx ¼
x
nþ1
n þ 1
, except n ¼1
8.
Z
f
0
ðxÞdx
f ðxÞ
¼ log f ðxÞ, ðdf ðxÞ¼f
0
ðxÞdxÞ
9.
Z
dx
x
¼ log x
10.
Z
f
0
ðxÞdx
2
ffiffiffiffiffiffiffiffiffi
f ðxÞ
p
¼
ffiffiffiffiffiffiffiffiffi
f ðxÞ
p
, ðdf ðxÞ¼f
0
ðxÞdxÞ
11.
Z
e
x
dx ¼ e
x
12.
Z
e
ax
dx ¼ e
ax
=a
13.
Z
b
ax
dx ¼
b
ax
a log b
, ðb > 0Þ
14.
Z
log xdx¼ x log x x
15.
Z
a
x
log adx¼ a
x
, ða > 0Þ
16.
Z
dx
a
2
þ x
2
¼
1
a
tan
1
x
a
17.
Z
dx
a
2
x
2
¼
1
a
tanh
1
x
a
or
1
2a
log
a þ x
a x
, ða
2
> x
2
Þ
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18.
Z
dx
x
2
a
2
¼
1
a
coth
1
x
a
or
1
2a
log
x a
x þ a
, ðx
2
> a
2
Þ
8
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A-20
INTEGRALS (Continued)
19.
Z
dx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
x
2
p
¼
sin
1
x
jaj
or
cos
1
x
jaj
, ða
2
> x
2
Þ
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20.
Z
dx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
a
2
p
¼ log ðx þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
a
2
p
Þ
21.
Z
dx
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
2
a
2
p
¼
1
jaj
sec
1
x
a
22.
Z
dx
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
x
2
p
¼
1
a
log
a þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
x
2
p
x
!
FORMS CONTAINING (a þbx)
For forms containing a þbx, but not listed in the table, the substitution u ¼
a þ bx
x
may prove helpful.
23.
Z
ða þ bxÞ
n
dx ¼
ða þ bxÞ
nþ1
ðn þ 1Þb
, ðn 6¼1Þ
24.
Z
xða þ bxÞ
n
dx ¼
1
b
2
ðn þ 2Þ
ða þ bxÞ
nþ2
a
b
2
ðn þ 1Þ
ða þ bxÞ
nþ1
, ðn 6¼1, 2Þ
25.
Z
x
2
ða þ bxÞ
n
dx ¼
1
b
3
ða þ bxÞ
nþ3
n þ 3
2a
ða þ bxÞ
nþ2
n þ 2
þ a
2
ða þ bxÞ
nþ1
n þ 1
26.
Z
x
m
ða þ bxÞ
n
dx ¼
x
mþ1
ða þ bxÞ
n
m þ n þ 1
þ
an
m þ n þ 1
Z
x
m
ða þ bxÞ
n1
dx
or
1
aðn þ 1Þ
x
mþ1
ða þ bxÞ
nþ1
þðm þn þ 2Þ
Z
x
m
ða þ bxÞ
nþ1
dx
or
1
bðm þ n þ 1Þ
x
m
ða þ bxÞ
nþ1
ma
Z
x
m1
ða þ bxÞ
n
dx
8
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27.
Z
dx
a þ bx
¼
1
b
log ða þ bxÞ
28.
Z
dx
ða þ bxÞ
2
¼
1
bða þ bxÞ
29.
Z
dx
ða þ bxÞ
3
¼
1
2bða þ bxÞ
2
30.
Z
xdx
a þ bx
¼
1
b
2
½a þ bx a log ða þ bxÞ
or
x
b
a
b
2
log ða þ bxÞ
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A-21
INTEGRALS (Continued)
31.
Z
xdx
ða þ bxÞ
2
¼
1
b
2
log ða þ bxÞþ
a
a þ bx
32.
Z
xdx
ða þ bxÞ
n
¼
1
b
2
1
ðn 2Þða þ bxÞ
n2
þ
a
ðn 1Þða þ bxÞ
n1
, n 6¼ 1, 2
33.
Z
x
2
dx
a þ bx
¼
1
b
3
1
2
ða þ bxÞ
2
2aða þ bxÞþa
2
log ða þ bxÞ
34.
Z
x
2
dx
ða þ bxÞ
2
¼
1
b
3
a þ bx 2a log ða þ bxÞ
a
2
a þ bx
35.
Z
x
2
dx
ða þ bxÞ
3
¼
1
b
3
log ða þ bxÞþ
2a
a þ bx
a
2
2ða þ bxÞ
2
36.
Z
x
2
dx
ða þ bxÞ
n
¼
1
b
3
1
ðn 3Þða þ bxÞ
n3
þ
2a
ðn 2Þða þ bxÞ
n2
a
2
ðn 1Þða þ bxÞ
n1
, n 6¼ 1, 2, 3
37.
Z
dx
xða þ bxÞ
¼
1
a
log
a þ bx
x
38.
Z
dx
xða þ bxÞ
2
¼
1
aða þ bxÞ
1
a
2
log
a þ bx
x
39.
Z
dx
xða þ bxÞ
3
¼
1
a
3
1
2
2a þ bx
a þ bx
2
þlog
x
a þ bx
"#
40.
Z
dx
x
2
ða þ bxÞ
¼
1
ax
þ
b
a
2
log
a þ bx
x
41.
Z
dx
x
3
ða þ bxÞ
¼
2bx a
2a
2
x
2
þ
b
2
a
3
log
x
a þ bx
42.
Z
dx
x
2
ða þ bxÞ
2
¼
a þ 2bx
a
2
xða þ bxÞ
þ
2b
a
3
log
a þ bx
x
FORMS CONTAINING c
2
x
2
, x
2
c
2
43.
Z
dx
c
2
þ x
2
¼
1
c
tan
1
x
c
44.
Z
dx
c
2
x
2
¼
1
2c
log
c þ x
c x
, ðc
2
> x
2
Þ
45.
Z
dx
x
2
c
2
¼
1
2c
log
x c
x þ c
, ðx
2
> c
2
Þ
46.
Z
xdx
c
2
x
2
¼
1
2
log ðc
2
x
2
Þ
47.
Z
xdx
ðc
2
x
2
Þ
nþ1
¼
1
2nðc
2
x
2
Þ
n
48.
Z
dx
ðc
2
x
2
Þ
n
¼
1
2c
2
ðn 1Þ
x
ðc
2
x
2
Þ
n1
þð2n 3Þ
Z
dx
ðc
2
x
2
Þ
n1
49.
Z
dx
ðx
2
c
2
Þ
n
¼
1
2c
2
ðn 1Þ
x
ðx
2
c
2
Þ
n1
ð2n 3Þ
Z
dx
ðx
2
c
2
Þ
n1
50.
Z
xdx
x
2
c
2
¼
1
2
log ðx
2
c
2
Þ
A-22
INTEGRALS (Continued)
51.
Z
xdx
ðx
2
c
2
Þ
nþ1
¼
1
2n ðx
2
c
2
Þ
n
FORMS CONTAINING a þbx and c þdx
u ¼a þbx, v ¼ c þ dx, k ¼ad bc
If k ¼0, then v ¼
c
a
u
52.
Z
dx
u v
¼
1
k
log
v
u
53.
Z
xdx
u v
¼
1
k
a
b
log ðuÞ
c
d
log ðvÞ
hi
54.
Z
dx
u
2
v
¼
1
k
1
u
þ
d
k
log
v
u
55.
Z
xdx
u
2
v
¼
a
bku
c
k
2
log
v
u
56.
Z
x
2
dx
u
2
v
¼
a
2
b
2
ku
þ
1
k
2
c
2
d
log ðvÞþ
aðk bcÞ
b
2
log ðuÞ
57.
Z
dx
u
n
v
m
¼
1
kðm 1Þ
1
u
n1
v
m1
ðm þn 2Þb
Z
dx
u
n
v
m1
58.
Z
u
v
dx ¼
bx
d
þ
k
d
2
log ðvÞ
59.
Z
u
m
dx
v
n
¼
1
kðn 1Þ
u
mþ1
v
n1
þ bðn m 2Þ
Z
u
m
v
n1
dx
or
1
dðn m 1Þ
u
m
v
n1
þ mk
Z
u
m1
v
n
dx
or
1
dðn 1Þ
u
m
v
n1
mb
Z
u
m1
v
n1
dx
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FORMS CONTAINING (a þbx
n
)
60.
Z
dx
a þ bx
2
¼
1
ffiffiffiffiffi
ab
p
tan
1
x
ffiffiffiffiffi
ab
p
a
, ðab > 0Þ
61.
Z
dx
a þ bx
2
¼
1
2
ffiffiffiffiffiffiffiffiffi
ab
p
log
a þ x
ffiffiffiffiffiffiffiffiffi
ab
p
a x
ffiffiffiffiffiffiffiffiffi
ab
p
, ðab < 0Þ
or
1
ffiffiffiffiffiffiffiffiffi
ab
p
tanh
1
x
ffiffiffiffiffiffiffiffiffi
ab
p
a
, ðab < 0Þ
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62.
Z
dx
a
2
þ b
2
x
2
¼
1
ab
tan
1
bx
a
63.
Z
xdx
a þ bx
2
¼
1
2b
logða þ bx
2
Þ
64.
Z
x
2
dx
a þ bx
2
¼
x
b
a
b
Z
dx
a þ bx
2
65.
Z
dx
ða þ bx
2
Þ
2
¼
x
2aða þ bx
2
Þ
þ
1
2a
Z
dx
a þ bx
2
A-23
INTEGRALS (Continued)
66.
Z
dx
a
2
b
2
x
2
¼
1
2ab
log
a þ bx
a bx
67.
Z
dx
ða þ bx
2
Þ
mþ1
¼
1
2ma
x
ða þ bx
2
Þ
m
þ
2m 1
2ma
Z
dx
ða þ bx
2
Þ
m
or
ð2mÞ!
ðm!Þ
2
x
2a
X
m
r¼1
r!ðr 1Þ!
ð4aÞ
mr
ð2rÞ!ða þ bx
2
Þ
r
þ
1
ð4aÞ
m
Z
dx
a þ bx
2
"#
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68.
Z
xdx
ða þ bx
2
Þ
mþ1
¼
1
2bmða þ bx
2
Þ
m
69.
Z
x
2
dx
ða þ bx
2
Þ
mþ1
¼
x
2mbða þ bx
2
Þ
m
þ
1
2mb
Z
dx
ða þ bx
2
Þ
m
70.
Z
dx
xða þ bx
2
Þ
¼
1
2a
log
x
2
a þ bx
2
71.
Z
dx
x
2
ða þ bx
2
Þ
¼
1
ax
b
a
Z
dx
a þ bx
2
72.
Z
dx
xða þ bx
2
Þ
mþ1
¼
1
2amða þ bx
2
Þ
m
þ
1
a
Z
dx
xða þ bx
2
Þ
m
or
1
2a
mþ1
X
m
r¼1
a
r
rða þ bx
2
Þ
r
þ log
x
2
a þ bx
2
"#
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73.
Z
dx
x
2
ða þ bx
2
Þ
mþ1
¼
1
a
Z
dx
x
2
ða þ bx
2
Þ
m
b
a
Z
dx
ða þ bx
2
Þ
mþ1
74.
Z
dx
a þ bx
3
¼
k
3a
1
2
log
ðk þ xÞ
3
a þ bx
3
þ
ffiffiffi
3
p
tan
1
2x k
k
ffiffiffi
3
p
, k ¼
ffiffiffi
a
b
3
r
75.
Z
xdx
a þ bx
3
¼
1
3bk
1
2
log
a þ bx
3
ðk þ xÞ
3
þ
ffiffiffi
3
p
tan
1
2x k
k
ffiffiffi
3
p
, k ¼
ffiffiffi
a
b
3
r
76.
Z
x
2
dx
a þ bx
3
¼
1
3b
logða þ bx
3
Þ
77.
Z
dx
a þ bx
4
¼
k
2a
1
2
log
x
2
þ 2kx þ 2k
2
x
2
2kx þ 2k
2
þ tan
1
2kx
2k
2
x
2
, ab > 0, k ¼
ffiffiffiffiffi
a
4b
4
r
78.
Z
dx
a þ bx
4
¼
k
2a
1
2
log
x þ k
x k
þ tan
1
x
k
, ab < 0, k ¼
ffiffiffiffiffiffiffi
a
b
4
r
79.
Z
xdx
a þ bx
4
¼
1
2bk
tan
1
x
2
k
, ab > 0, k ¼
ffiffiffi
a
b
r
80.
Z
xdx
a þ bx
4
¼
1
4bk
log
x
2
k
x
2
þ k
, ab < 0, k ¼
ffiffiffiffiffiffiffi
a
b
r
81.
Z
x
2
dx
a þ bx
4
¼
1
4bk
1
2
log
x
2
2kx þ 2k
2
x
2
þ 2kx þ 2k
2
þ tan
1
2kx
2k
2
x
2
, ab > 0, k ¼
ffiffiffiffiffi
a
4b
4
r
A-24