1 Techniques of integration
a substitution
use when the integral contains a function
and its derivative
Example
1 1 2 1 dx
Ix city
dx fu du
4 C
x2 1 c
f
raytigf.ee
I 2 0
312413
C Partial fractions
examp
I
5 12 Ax B x 5 1 2 4
15 12 Ax 5Ax BX 5 B Cx 4c
15 12 Ax x2 Bx 5Ax 4C 513
5 12 Atc B 5A 41 5 B
A 3
41 5B 12 5B 4 A 12
IEE.IE
a
II
5
, Iiii s 4 Is
PracticeProblem
Six In ax si 2 3 1
In 21 Fenixtil c
e In 2s In
EE ii i
75 3 a
3
3331113
A
9 13
d Integration by Parts
formula
fudu uv fudu
Practice problems
fxetdx
Y
xe
ax
f e dx
xe etc
trigonometric Integrals
Theseinvolveintegrals thatinclude sinx cosx tan x seex
ee
E.EE Ii
cos2x 1 osczx
2
, use identities to reduce even powers
If powers are odd take oneout and use
identities to simplify
Example sins cosdx fu du
c sin c
u sinx
du costdx
PracticeProblem sin cos x dx 1 10312x 1 x
02
1 2x 1 1 02147
q 4
1 103 12 cost dx Cos 4
4
fx cos tx dx
f sin ax c
Trig substitution
use for integralsinvolving squareroots
Fx asino
2 Ear x a sect
3 Tax a ton
Example
Ex a
2sint dx 2costdo
costdo
teasing
cos to
a e
sin't cos'o 1
cos'o 1 sin't
50do I do C
21
ÉE 5 E C
, IE bTE
PracticeProblem
ax
Sx a
ÉÉsÉÉantdo
3sec tent do
19sec't
vÉoa
3sec tone do
9sec o É
psecto sent
Betendo
do cos do
age
g
sint c
Ff c
3sec
V4
sect
cos
it
a
Improper Integrals
Linef
2
f dx x
time x 1
converges
o
2 ax timef.tt timeIn
diverges
a substitution
use when the integral contains a function
and its derivative
Example
1 1 2 1 dx
Ix city
dx fu du
4 C
x2 1 c
f
raytigf.ee
I 2 0
312413
C Partial fractions
examp
I
5 12 Ax B x 5 1 2 4
15 12 Ax 5Ax BX 5 B Cx 4c
15 12 Ax x2 Bx 5Ax 4C 513
5 12 Atc B 5A 41 5 B
A 3
41 5B 12 5B 4 A 12
IEE.IE
a
II
5
, Iiii s 4 Is
PracticeProblem
Six In ax si 2 3 1
In 21 Fenixtil c
e In 2s In
EE ii i
75 3 a
3
3331113
A
9 13
d Integration by Parts
formula
fudu uv fudu
Practice problems
fxetdx
Y
xe
ax
f e dx
xe etc
trigonometric Integrals
Theseinvolveintegrals thatinclude sinx cosx tan x seex
ee
E.EE Ii
cos2x 1 osczx
2
, use identities to reduce even powers
If powers are odd take oneout and use
identities to simplify
Example sins cosdx fu du
c sin c
u sinx
du costdx
PracticeProblem sin cos x dx 1 10312x 1 x
02
1 2x 1 1 02147
q 4
1 103 12 cost dx Cos 4
4
fx cos tx dx
f sin ax c
Trig substitution
use for integralsinvolving squareroots
Fx asino
2 Ear x a sect
3 Tax a ton
Example
Ex a
2sint dx 2costdo
costdo
teasing
cos to
a e
sin't cos'o 1
cos'o 1 sin't
50do I do C
21
ÉE 5 E C
, IE bTE
PracticeProblem
ax
Sx a
ÉÉsÉÉantdo
3sec tent do
19sec't
vÉoa
3sec tone do
9sec o É
psecto sent
Betendo
do cos do
age
g
sint c
Ff c
3sec
V4
sect
cos
it
a
Improper Integrals
Linef
2
f dx x
time x 1
converges
o
2 ax timef.tt timeIn
diverges
