Question

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

Answer 1

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

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

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

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

$2\int (\cos x-\sin x)(\cos x+\sin x)dx$

$\left[\right(a-b\left)\right(a+b)=a^2-b^2]$

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

$2\int \cos 2xdx$

$2\frac{\sin x}{2}+C$

$\sin x+C$ (Ans)