Mark the correct alternative in each of the following:
Evaluate $\int \frac{\sin x}{3+4 \cos ^{2} x} d x$
A. $\log \left(3+4 \cos ^{x} x\right)+C$
B. $\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{\cos x}{\sqrt{3}}\right)+C$
C. $-\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos \mathrm{x}}{\sqrt{3}}\right)+\mathrm{C}$
D. $\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos x}{\sqrt{3}}\right)+C$
$\int \frac{\sin x}{3+4(\cos x)^{2}} d x$
$\Rightarrow \cos x=t$ then ;
$\Rightarrow-\sin (x) d x=d t$
$=-\int \frac{d t}{3+4 t^{2}}\left(\int \frac{d t}{a+b t^{2}}=\frac{1}{\sqrt{a b}} \tan ^{-1} \sqrt{\frac{b}{a}}\right)$
$=-\frac{1}{2 \sqrt{3}} \tan ^{-1} \sqrt{\frac{4}{3}} t$ put $(\cos x=t)$
$\Rightarrow-\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos x}{\sqrt{3}}\right)+C$