Question

What is the integral of `int cos^3 (x)dx` from 0 to pi/2?

Answer 1

`int_0^(pi/2) cos^3xdx = 2/3`

Explanation:

Use the idendity `cos^2x+sin^2x = 1`:

`int_0^(pi/2) cos^3xdx = int_0^(pi/2) cosx cos^2xdx = int_0^(pi/2) cosx (1-sin^2x) xdx`

substitute:

`t=sinx`
`dt = cosx dx`
`x in (0,pi/2) => t in (0,1)`

so:

`int_0^(pi/2) cos^3xdx = int_0^1 (1-t^2)dt = [t-t^3/3]_0^1 = 2/3`