Integrate (Sinx)^2

Prove It

MHF Helper
Aug 2008
12,883
4,999
Hi!

How do I integrate (Sinx)^2 ?
Recall that \(\displaystyle \cos^2{x} + \sin^2{x} = 1\) and \(\displaystyle \cos^2{x} - \sin^2{x} = \cos{2x}\).

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

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

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

\(\displaystyle \sin^2{x} = \frac{1}{2} - \frac{1}{2}\cos{2x}\).



Therefore:

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

\(\displaystyle = \frac{1}{2}x - \frac{1}{4}\sin{2x} + C\).
 
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