Integrals

gistfile1.txt

$\int{f(x) g'(x) \ dx} = fg - \int{ g(x) f'(x) \ dx}$ 

$\int{cos^2x \ dx}  = \int{cos \ x \ cos \ x \ dx}  = \int{cos \ x  \ (sin \ x)' \ dx}  = cos \ x  \ sin \ x - \int{sin \ x  \ (cos \ x)' \ dx}  = cos \ x  \ sin \ x - \int{sin \ x  \ sin \ x \ dx}  =  cos \ x  \ sin \ x - \int{sin^2 \ x \ dx}  =  sin\ x  \ cos \ x + \int{sin^2 \ x \ dx}  =  sin\ x  \ cos \ x + \int{ (1 - cos^2 \ x) \ x \ dx} =  sin\ x  \ cos \ x + \int{ 1 \ dx} - \int{ cos^2 \ x \ dx} = sin\ x  \ cos \ x + x - \int{ cos^2 \ x \ dx}  =>  \int \cos ^2x\,dx=\frac{x+\sin x\cos x}{2}+C. $