How do you integrate ##int sin^2(x) dx## using integration by parts?

##intsin^2(x)dx=x/2-sin(2x)/4+c##, where ##c## is the constant of integration.

##intsin^2(x)dx=intfrac{d}{dx}(x)sin^2(x)dx##

##=xsin^2(x)-intxfrac{d}{dx}(sin^2(x))dx##

##=xsin^2(x)-intx(2sin(x)cos(x))dx##

##=xsin^2(x)-intxsin(2x)dx##

##=xsin^2(x)+1/2intxfrac{d}{dx}(cos(2x))dx##

##=xsin^2(x)+1/2[xcos(2x)-intfrac{d}{dx}(x)cos(2x)dx]##

##=xsin^2(x)+1/2xcos(2x)-1/2intcos(2x)dx##

##=xsin^2(x)+1/2xcos(2x)-sin(2x)/4+c##, where ##c## is the constant of integration.

##=x/2(cos(2x)+2sin^2(x))-sin(2x)/4+c##

##=x/2-sin(2x)/4+c##