1. Integrals Involving Powers of Sine and Cosine

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

9. 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

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 cos2 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=cos4x
du= -4sin4x 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 sec2 x or tan x sec x

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

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