1. MTH 252 Integral Calculus Chapter 8 – Principles of
Integral Evaluation
Section 8.3 – Trigonometric
Integrals
Copyright © 2005 by Ron Wallace, all rights reserved.

2. Odd Powers of SIN & COS n > 2 is a positive odd integer
n - 1 is a positive even integer
Multiply out the polynomial, integrate, and substitute back.

3. Odd Powers of SIN & COS n > 2 is a positive odd integer
n - 1 is a positive even integer
Multiply out the polynomial, integrate, and substitute back.

4. Odd Powers of SIN & COS m is a positive integer
n is a positive odd integer
n - 1 is a positive even integer
Multiply out the polynomial, integrate, and substitute back.

5. Odd Powers of SIN & COS m is a positive odd integer
n is a positive integer
m - 1 is a positive even integer
Multiply out the polynomial, integrate, and substitute back.

6. Powers of SIN & COS
NOTE: m & n are non-negative integers.
If n is odd, put one of the cosines w/ dx, change the remaining cosines to sines, and let u = sin x.
If m is odd, put one of the sines w/ dx, change the remaining sines to cosines, and let u = cos x.
All other cases … use some other method.

7. Powers of TAN & SEC
For what values of m & n can the same approach be used?
[NOTE: m & n are non-negative integers.]
If m is odd and n > 0, put one of the tangents and one of the secants w/ dx, change the remaining tangents to secants, and let u = sec x.
If n is even and n > 0, put two of the secants w/ dx, change the remaining secants to tangents, and let u = tan x.
All other cases … use some other method.

8. Even Powers of SIN & COS m and n are BOTH non-negative even integers.
Remember the half-angle formulas:

9. Even Powers of SIN & COS Use the trigonometric identities …
m and n are BOTH non-negative even integers.
Use the trigonometric identities …
Multiply everything out.
Integrate each term, one at a time.

10. Powers of SEC
Memorize this one!

11. Use Integration by Parts
Powers of SEC
Use Integration by Parts
n > 2 and a positive integer

12. Powers of SEC Use Integration by Parts n > 2 and a positive integer
This kind of identity is called a “Reduction Formula.”
No! You do NOT need to memorize this one. Just use it.

13. Powers of TAN
Memorize this one too!

14. n > 1 and a positive integer
Powers of TAN
n > 1 and a positive integer
If n is even, change to secants using …
If n is odd,
Put sec x tan x w/ dx as follows …
Convert all tangents (except the one w/ dx) to secants.
Use the substitution, u = sec x

15. FYI: More Reduction Formulas

16. An equivalent form to the other solution.
One more note …
An equivalent form to the other solution.