How do I integrate sin^2(x)?

First, remember the compound angle formula for cosine:

cos(2x)=cos^2(x)-sin^2(x).  Now use the identity sin^2(x)+cos^2(x)=1 to give:

cos(2x)=(1-sin^2(x))-sin^2(x)=1-2sin^2(x)

Rearranging this so we have sin^2(x)=1/2(1-cos(2x))

Replace this with the original integration and use the chain rule to get:

1/2(x-1/2sin(2x))+c