The main purpose of trig integrals is getting a setup like the usual u-sub problem:
$$
∫f(g(x))g'(x)dx
$$
where you are looking for a trig function and its derivative
General Notes:
When a trig function has an odd power:
- split off one factor of the function
- lone factor acts as the derivative piece for u-sub
- even powered factor can be rewritten using identities
- Trig identities (half angle, pythagorean, etc) just let us rewrite everything as one trig function
- If both powers of trig functions are even, use half-angle identities
- Half angle identities lower the power and turn squares into linear trig expressions, which we then can take integrals of
| Situation |
Strat |
| Odd power |
derivative matching / pythagorean identity on remaining even power |
| Both even |
half-angle identities |
Example:
$$
∫sin^3xcos^2x dx
$$
- we know the derivative of cosine is -sine, so we let u=cosx
- We have $sin^3(x)$ which we can write as $sin^2xsinx$
- A single $sinxdx$ allows for derivative matching, and the remaining $sin^2x$ can be converted into Pythagorean theorem $sin^2x=1-cos^2x$
$$
∫sin^2xsinx cos^2xdx
$$
- Convert leftover sine into the identity
$$
∫(1-cos^2x)sinxcos^2xdx
$$
- Now we can use u=cosx and du=sinx dx as we see the derivative