1. More Trigonometric Integrals
Lesson 9.4

2. Recall Basic Identities
Pythagorean Identities
Half-Angle Formulas
These will be used to integrate powers of sin and cos

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

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

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

6. 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

7. Combinations of sin, cos
Consider
Use Pythagorean identity
Separate and use sinn x strategy for n odd

8. Combinations of tanm, secn
When n is even
Factor out sec2 x
Rewrite remainder of integrand in terms of Pythagorean identity sec2 x = 1 + tan2 x
Then u = tan x, du = sec2x dx
Try

9. Combinations of tanm, secn
When m is odd
Factor out tan x sec x (for the du)
Use identity sec2 x – 1 = tan2 x for even powers of tan x
Let u = sec x, du = sec x tan x
Try the same integral with this strategy
Note similar strategies for integrals involving combinations of cotm x and cscn x

10. Integrals of Even Powers of sec, csc
Use the identity sec2 x – 1 = tan2 x
Try

11. Assignment
Lesson 9.4
Page 376
Exercises1 – 33 odd