Question

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

Answer 1

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

$=\cos 2\theta \int \frac{dx}{{\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x}$

$=\cos 2\theta \int \frac{dx}{1+\sin 2x}$

$=\cos 2\theta \int \frac{1-\sin 2x}{(1+\sin 2x)(1-\sin 2x)}dx$

$=\cos 2\theta \int \frac{1-\sin 2x}{1-{\sin }^{2}2x}dx$

$=\cos 2\theta \int \frac{1-\sin 2x}{{\cos }^{2}2x}dx$

$=\cos 2\theta (\int {\sec }^{2}2xdx-\int \sec x\tan xdx)$

$=\cos 2\theta (\frac{\tan 2x}{2}-\frac{\sec 2x}{2}+c)$

$=\frac{\cos 2\theta }{2}(\tan 2x-\sec 2x+c)$