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3 Integrals Involving Powers of Sine and Cosine

4 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.

5 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  sin m x cos n x dx into forms to which you can apply the Power Rule, use the following identities.

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

7 Integrals Involving Powers of Sine and Cosine

8 Integral of sin n 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

9 Integral of sin n x, n Odd  Integrate and un-substitute  Similar strategy with cos n x, n odd

10 Integral of sin n x, n Even  Use half-angle formulas  Try Change to power of cos 2 x  Expand the binomial, then integrate

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

12 Try with Let u=sinx du=cosx dx

13 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.

14 Example – Solution cont’d

15 Combinations of sin, cos  Consider  Use Pythagorean identity

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

17 Integrals Involving Powers of Secant and Tangent

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

19 Integrals Involving Powers of Secant and Tangent cont’d

20 Example 4 – Power of Tangent Is Odd and Positive Find Solution: Because you expect to use the Power Rule with u = sec x, save a factor of (sec x tan x) to form du and convert the remaining tangent factors to secants.

21 Example 4 – Solution cont’d

22 Combinations of tan m, sec n  Try factoring out sec 2 x or tan x sec x

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

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

25 Wallis's Formulas  If n is odd and (n ≥ 3) then  If n is even and (n ≥ 2) then And … Believe it or not These formulas are also valid if cos n x is replaced by sin n x And … Believe it or not These formulas are also valid if cos n x is replaced by sin n x

26 Wallis's Formulas  Try it out …

27 These formulas are also valid if cos n x is replaced by sin n x. Integrals Involving Powers of Sine and Cosine

28 Integrals Involving Sine-Cosine Products with Different Angles

29 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.

30 Example 8 – Using Product-to-Sum Identities Find Solution: Considering the second product-to-sum identity above, you can write