Question

What is the antiderivative of `(sin(x))^2`?

Answer 1

`= 1/2 (x -1/2 sin 2x) + C`

Explanation:

use the handy double-angle formula

`cos 2A = cos^2 A - sin^2 A`
`= 2 cos^2 A -1`
`= 1 - 2 sin^2 A`

last one gives us `sin^2 A = 1/2(1- cos 2A)`

So

`int (sin(x))^2 dx`

`= 1/2 int 1 - cos 2x dx`

`= 1/2 (x -1/2 sin 2x) + C`

which you can flip using the other double-angle formula: `sin 2A = 2 sin A cos A`

`= 1/2 (x - sin x cos x) + C`