Mark the correct alternative in each of the following:

Question:

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. $\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos x}{\sqrt{3}}\right)+C$

Solution:

$\int \frac{\sin x}{3+4(\cos x)^{2}} d x$

$\Rightarrow \cos x=t$ 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 $(\cos x=t)$

$\Rightarrow-\frac{1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos x}{\sqrt{3}}\right)+C$

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