# Integrate (Sinx)^2

• May 22nd 2010, 12:26 AM
Fredrik373
Integrate (Sinx)^2
Hi!

How do I integrate (Sinx)^2 ?
• May 22nd 2010, 12:41 AM
Prove It
Quote:

Originally Posted by Fredrik373
Hi!

How do I integrate (Sinx)^2 ?

Recall that $\cos^2{x} + \sin^2{x} = 1$ and $\cos^2{x} - \sin^2{x} = \cos{2x}$.

So if you subtract the second equation from the first...

$(\cos^2{x} + \sin^2{x}) - (\cos^2{x} - \sin^2{x}) = 1 - \cos{2x}$

$2\sin^2{x} = 1 - \cos{2x}$

$\sin^2{x} = \frac{1}{2} - \frac{1}{2}\cos{2x}$.

Therefore:

$\int{\sin^2{x}\,dx} = \int{\left(\frac{1}{2} - \frac{1}{2}\cos{2x}\right)\,dx}$

$= \frac{1}{2}x - \frac{1}{4}\sin{2x} + C$.