Question

Integrate

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

• A. $\sin 2x+c$
• B. $\cos 2x+c$
• C. $\tan 2x+c$
• D. $\cot 2x+c$

Answer 1

The correct option is A $\sin 2x+c$

Let $y=\int \left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)(2+2\sin 2x)dx$

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

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

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

$=\int 2\cos 2xdx$

$\therefore y=\sin 2x+c$

$\therefore \frac{dy}{dx}=\frac{d}{dx}(\sin 2x+c)=2\cos 2x$