Question

Evaluate $\int \frac{\cos 2x}{(\sin x+\cos x{)}^2}dx$

Answer 1

$I=\int \frac{\cos 2x}{(\sin x+\cos x{)}^2}dx$

$I=\int \frac{co{s}^{2}x-si{n}^{2}x}{(sinx+cosx{)}^2}dx$

$I=\int \frac{(sinx+cosx)(cosx-sinx)}{(sinx+cosx{)}^2}dx$

$I=\int \frac{cosx-sinx}{cosx+sinx}dx$

Let $cosx+sinx=t$

$(cosx-sinx)dx=dt$

$I=\int \frac{dt}{t}$

$I=log|t|+C$

$I=log|cosx+sinx|+C$