Download Integrals Involving Powers of Sine and Cosine

AP CALCULUS BC
Section Number:
LECTURE NOTES
Topics: Trigonometric Integrals
MR. RECORD
Day: 1 of 1
8.3
Integrals Involving Powers of Sine and Cosine
In this section you will study techniques for evaluating integrals of the form
m
n
m
n
 sin x cos x dx and  sec x tan x dx
where either m or n is a positive integer.
The trick to finding antiderivatives of these forms is to break them into combinations of trigonometric integrals
to which you can apply the Power Rule.
This can be accomplished by using the following trigonometric identities:
Example 1: Power of Sine is Odd and Positive.
Evaluate  sin 3 x cos4 x dx
Example 2: Power of Cosine is Odd and Positive.

Evaluate
3


6
cos 3 x
dx
sin x
Example 3: Power of Cosine is Even and Nonnegative.
Evaluate  cos 4 x dx
Wallis’s Formulas
1. If n is odd  n  3 , then

 2  4  6   n  1 
x dx      
.
 3  5  7   n 
0
2. If n is even  n  2  , then
2
 cos

n
2
 cos
0
n
 1  3  5 
x dx     
 2  4  6 
 n  1   

  .
 n  2 
Note: The above formulas are also valid if cosn x is replaced by sin n x .
Integrals Involving Powers of Secant and Tangent
Example 4: Power of Tangent is Odd and Positive.
Evaluate
tan 3 x
 sec x dx
Example 5: Power of Secant is Even and Positive.
Evaluate  sec4 3x tan 3 3x dx
Example 6: Power of Tangent is Even.

Evaluate
4
 tan
4
x dx
0
Sometimes, none of the rules seem to apply.
Example 7: Converting to Sines and Cosines.
Evaluate
sec x
dx
2
x
 tan
Integrals Involving Sine-Cosine Products with Different Angles
Integrals involving the products of sines and cosines of two different angles occur in many applications. In such
instances you can use the following product-to-sum identities.
Example 8: Using Product-to-Sum Identities.
Evaluate  sin 5x cos 4 xdx