Integrals Involving Powers of Sine and Cosine

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.

Basic Identities Pythagorean Identities Half-Angle Formulas These will be used to integrate powers of sine and cosine Basic Identities Pythagorean Identities Half-Angle Formulas

Integrals Involving Powers of Sine and Cosine

Try with Let u=sinx du=cosx dx

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.

Example – Solution cont’d

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.

Integral of sinn x, n Odd Split into product of an even power and sin x Make the even power a power of sin2 x Use the Pythagorean identity Let u = cos x, du = -sin x dx

Integral of sinn x, n Odd Integrate and un-substitute Similar strategy with cosn x, n odd

Integral of sinn x, n Even Use half-angle formulas Try Change to power of cos2 x Expand the binomial, then integrate

Combinations of sin, cos Try with General form 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 If the powers of both sine and cosine are even, use the power reducing formulas:

Combinations of sin, cos Consider Use Pythagorean identity

Combinations of sin, cos u=cos4x du= -4sin4x dx

Integrals Involving Powers of Secant and Tangent

Integrals Involving Powers of Secant and Tangent The following guidelines can help you evaluate integrals of the form

Integrals Involving Powers of Secant and Tangent cont’d

Combinations of tanm, secn Try factoring out sec2 x or tan x sec x

Integrals of Even Powers of sec, csc Use the identity sec2 x – 1 = tan2 x Try u=tan3x du=3sec^2(3x) dx

Homework Section 8.3, pg. 540: 5-17 odd, 25-29 odd