Question

Evaluate :

$\int \frac{\sin x}{\sqrt{3+2\cos x}}dx$

• A. $-\sqrt{3+2\cos x}+C$
• B. $-2+\sqrt{\cos x}+C$
• C. $-\sqrt{2+3\cos x}+C$
• D. None of these

Answer 1

The correct option is A $-\sqrt{3+2\cos x}+C$

$I=\int \frac{\sin x}{\sqrt{3+2\cos x}}dx$

Let $3+2\cos x=t$.

Then, $d(3+2\cos x)=dt$

$-2\sin xdx=dt$

$dx=-\frac{dt}{2\sin x}$

Putting $3+2\cos x=t$ and $dx=-\frac{dt}{2\sin x}$, we get,

$I=\int \frac{\sin x}{\sqrt{t}}\times -\frac{dt}{2\sin x}$

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

$I=-\frac{1}{2}\times \frac{t^{1/2}}{1/2}+C$

$I=-\sqrt{t}+C$

$I=-\sqrt{3+2\cos x}+C$