Question

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

Answer 1

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

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

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

Let $\cos x+\sin x=t$ $\Rightarrow (\cos x-\sin x)dx=dt$

$\Rightarrow I=\frac{dt}{t}$ $=\log \mid t\mid +c$

$\Rightarrow I=\log |\sin x+\cos x|+c$