7.2 Trigonometric Integrals

1. 7.2 Trigonometric Integrals In this section, we introduce techniques for evaluating integrals of the form where either m or n is a nonnegative integer and occasionally, integrals involving the product of sines and cosines of two different angles. REFER to the Handout for Trigonometric Integrals

2. Integrals Involving Sine and Cosine A. • If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosine. Then, expand and integrate. convert to cosine save for du odd

3. If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sine. Then, expand and integrate. convert to sine save for du odd • If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities • to convert the integrand to odd powers of the cosine. Then, proceed as in case 2. • It is sometimes helpful to use the identity and

4. Examples: Integrals Involving Sine and Cosine Evaluate the integral. 1) 2)

5. 3) 4)

6. Integrals Involving Secants and Tangents B. • If the power of the tangent is odd and positive, save a secant-tangent factor and convert the remaining factors to secants. Then, expand and integrate. convert to secants save for du odd

7. If the power of the secant is even and positive, save a secant squared factor and convert the remaining factors to tangents. Then, expand and integrate. convert to tangents save for du even • If there are no secant factors and the power of the tangent is even and positive, convert a tangent squared factor to secants; then expand and repeat if necessary. convert to secants

8. 4. If the integral is of the form , where n is odd and positive, possibly use integration by parts. 5. If none of the first four cases apply, try converting to sines and cosines.

9. Examples:Integrals Involving Secants and Tangents Evalute the integral. 1) Power Rule 2) Power Rule

10. 3) • Since the first two cases do not apply, try converting to sines and cosines.

11. Integrals Involving Cotangent and Cosecant NOTE: The guidelines for integrals involving cotangent and cosecant would be similar to that of integrals involving tangent and secant. Refer to your handout. Example:

12. Integrals Involving Sine-Cosine Products with Different angles In such instances, we use the following product-to-sum identities. Example: