Question

Integrate with respect to x:$\frac{\cos 2x}{{\left(\cos x+\sin x\right)}^2}$

Answer 1

Given that

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

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

We know that

${\sin }^{2}x+{\cos }^{2}x=1$

$\sin 2x=2\sin x\cos x$

Therefore,

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

Let $t=1+\sin 2x$

$\frac{dt}{dx}=0+2\cos 2x$

$\frac{dt}{2}=\cos 2xdx$

Therefore,

$I=\frac{1}{2}\int \frac{1}{t}dt$

$I=\frac{1}{2}\ln \left(t\right)+C$

On putting the value of $t$, we get

$I=\frac{1}{2}\ln (1+\sin 2x)+C$

Hence, this is the answer.