Question

integral of cos2x/(cosx+sinx)^2 dx

Answer 1

$\int \frac{ \cos2x}{ {( \cos x + \sin x)}^{2} } dx \\ = \int \frac{ \cos2x}{ 1 + \sin2x } dx \\ put \: \:1 + \sin2x = u \\ then \: \: \cos2x \: dx = \frac{du}{2} \\ integral \: \: becomes \\ \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} ln(u) + c \\ = ln( \sqrt{u} ) + c = ln( \sqrt{1 + \sin2x } ) + c$