7.2 Integration Techniques

Trigonometric Integrals

Question #2 (Medium): Evaluating the Integral of Sine & Cosine With Even Power

Strategy

When and or are raised to even power, half-angle identities are helpful in reducing the even

power by half every time it is applied. Then the reduced power, in conjunction with trigonometric

identities, the integral can be simplified and evaluated.

Sample Question

Evaluate the integral.

Solution

Since both and are raised to even power, no single term can be isolated. By applying half-

angle identities, even power is reduced by half every time it is applied. So here:

Then the three components in the integral can be evaluated separately.

So:

First:

Second:

. Let , then

Substituting into the integral,

Last component:

.

Put all three pieces together:

Therefore, the integral is evaluated as:

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###### Document Summary

Question #2 (medium): evaluating the integral of sine & cosine with even power. When and or are raised to even power, half-angle identities are helpful in reducing the even power by half every time it is applied. Then the reduced power, in conjunction with trigonometric identities, the integral can be simplified and evaluated. Since both and are raised to even power, no single term can be isolated. By applying half- angle identities, even power is reduced by half every time it is applied. Then the three components in the integral can be evaluated separately.

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