Integrals Involving Powers of and Cosine Sine 2

Integrals Involving Powers of and Cosine Sine 2

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. 3

Basic Identities These will be used to integrate powers of sine and cosine § Pythagorean Identities § Half-Angle Formulas 4

Integrals Involving Powers of Sine and Cosine 5

Try with Let u=sinx du=cosx dx 6

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. 7

Example – Solution cont’d 8

Integrals Involving Powers of Sine and Cosine For instance, you can evaluate sin 5 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. 9

Integral of sinn x, n Odd § Split into product of an even power and sin x § Make the even power a power of sin 2 x § Use the Pythagorean identity § Let u = cos x, du = -sin x dx 10

Integral of sinn x, n Odd § Integrate and un-substitute § Similar strategy with cosn x, n odd 11

Integral of sinn x, n Even § Use half-angle formulas § Try Change to power of cos 2 x § Expand the binomial, then integrate 12

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: 13

Combinations of sin, cos § Consider § Use Pythagorean identity 14

Combinations of sin, cos u=cos 4 x du= -4 sin 4 x dx 15

Integrals Involving Powers of Secant and Tangent 16

Integrals Involving Powers of Secant and Tangent The following guidelines can help you evaluate integrals of the form 17

Integrals Involving Powers of Secant and Tangent cont’d 18

Combinations of tanm, secn § Try factoring out sec 2 x or tan x sec x 21

Integrals of Even Powers of sec, csc § Use the identity sec 2 x – 1 = tan 2 x § Try u=tan 3 x du=3 sec^2(3 x) dx 22

Homework § Section 8. 3, pg. 540: 5 -17 odd, 25 -29 odd 23