8
Integration Techniques, L’Hôpital’s Rule,
and Improper Integrals
Copyright © Cengage Learning. All rights reserved.
Integrals Involving Powers of
Sine and Cosine
3
Integrals Involving Powers of Sine and Cosine
In this section you will study techniques for
evaluating integrals of the form
where either m or n is a positive integer.
To find antiderivatives for these forms, try to break
them into combinations of trigonometric integrals
to which you can apply the Power Rule.
4
Integrals Involving Powers of Sine and Cosine
For instance, you can evaluate  sin5 x cos x dx with the
Power Rule by letting u = sin x. Then, du = cos x dx and
you have
To break up  sinm x cosn x dx into forms to which you can
apply the Power Rule, use the following identities.
5
These will be used
to integrate powers
of sine and cosine
Basic Identities
 Pythagorean Identities
sin   cos   1
2
2
tan   1  sec 
2
2
1  cot   csc 
2
 Half-Angle Formulas
sin  
2
2
1  cos 2
2
cos  
2
1  cos 2
2
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Integrals Involving Powers of Sine and Cosine
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Integral of sinn x, n Odd
 Split into product of an even power and sin x
 sin
5
x dx 
 sin
4
x  sin x dx
 Make the even power a power of sin2 x
 sin
4
x  sin x dx 
  sin x 
2
2
sin x dx
 Use the Pythagorean identity
  sin x 
2
2
sin x dx 
 1  cos x 
2
2
sin x dx
 Let u = cos x, du = -sin x dx
  1  u
2

2
du    1  2 u  u du  ...
2
4
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Integral of sinn x, n Odd
 Integrate and un-substitute
  1  2 u  u du   u 
2
  cos x 
4
2
3
cos x 
3
1
2
u 
3
1
3
u C
5
5
cos x  C
5
5
 Similar strategy with cosn x, n odd
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Integral of sinn x, n Even
 Use half-angle formulas
cos  
2
1  cos 2
2
 Try
 cos 5 x dx
4
  cos
2
5x
Change to power of cos2 x
2
2
1

dx    1  cos 10 x   dx
2

 Expand the binomial, then integrate
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Combinations of sin, cos
Try with
 General form
 sin
m
x  cos xsin
dx2 x  cos 3 x dx

n
 If either n or m is odd, use techniques as before
– Split the odd power into an even power and power of
one
– Use Pythagorean identity
– Specify u and du, substitute
– Usually reduces to a polynomial
– Integrate, un-substitute
sin  
2
 If the powers of both sine and cosine are
 even, use the power reducing formulas:
1  cos 2
2
cos  
2
1  cos 2
2
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Try with
 sin x  cos x dx
2
 sin x  cos x dx 
2
3
 sin x cos x cos x dx
2
 sin x cos x cos x dx 
2
2
2
 u  u
2
4
2
 sin x (1  sin x ) cos xdx
 sin x (1  sin x ) cos xdx 
2
3
 du 
2
2
 (sin x  sin x ) cos xdx
2
4
 u du   u du
2
4
3
u
2
du   u du 
4
Let u=sinx
du=cosx dx
sin x
3
5

sin x
C
5
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Example – Power of Sine Is Odd and Positive
Find
Solution:
Because you expect to use the Power Rule with u = cos x,
save one sine factor to form du and convert the remaining
sine factors to cosines.
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Example – Solution
cont’d
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Combinations of sin, cos
 sin 4 x  cos 4 x dx
3
 Consider
2
 Use Pythagorean identity
 sin
2
4 x  cos 4 x  sin 4 x dx
2

 1  cos 4 x   cos 4 x  sin 4 x dx

  cos
2
2
2
4 x  cos 4 x   sin 4 x dx
4
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Combinations of sin, cos
u=cos4x
du= -4sin4x dx

  cos

1
4
2
4 x  cos 4 x   sin 4 x dx
4
 u  u
2
4
 du

1
12
u 
3
1
u c
5
20
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Integrals Involving Powers of
Secant and Tangent
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Integrals Involving Powers of Secant and Tangent
The following guidelines can help you evaluate integrals of
the form
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Integrals Involving Powers of Secant and Tangent
cont’d
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Combinations of tanm, secn
 Try factoring out sec2 x or tan x sec x
sec
y

tan
y
dy

4
3
22
Integrals of Even Powers of sec, csc
 Use the identity sec2 x – 1 = tan2 x
 Try
 sec 3 x dx
4
 sec 3 x  sec 3 x dx 
2
 1  tan

1
3
2
2
tan 3 x 
3 x  sec 3 x dx 
2
1
u=tan3x
du=3sec^2(3x) dx
tan 3 x  C
3
9
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Homework
 Section 8.3, pg. 540: 5-17 odd, 25-29 odd
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