Question

How do you find the integral of `int cos^3(x)*sin^6(x) dx`?

Answer 1

This belongs in your mathematical toolbox: `int sin^m u cos^n u du` with at least one of `m, n` odd.

Explanation:

`int sin^m u cos^n u du` with at least one of `m, n` odd.
Integrate by substitution. Do this by pulling off one from the odd power, then convert the remaining even power to the other function. Integrate the resulting polynomial in `sinu` or `cosu` term by term.

`I = int cos^3xsin^6x dx = int cos^2xsin^6x cosxdx`

`= int (1-sin^2x)sin^6x cosxdx`

`= int (sin^6x-sin^8x) cosxdx`

`= sin^7x/7 - sin^9/9x +C`