$\begin{aligned} & I=\int \frac{\sin x}{3+4 \cos ^2 x} d x \\ & \text { Put } \cos x=t \Rightarrow-\sin x d x=d t \\ & =\frac{-1}{4} \int \frac{d t}{t^2+\binom{\sqrt{3}}{2}^2} \\ & I=-\frac{1}{\text { 4. } \frac{\sqrt{3}}{2}} \cdot \tan ^{-1} \frac{t}{\left(\frac{\sqrt{3}}{2}\right)}+c=\frac{-1}{2 \sqrt{3}} \tan ^{-1} \frac{2 t}{\sqrt{3}}+c \\ & \Rightarrow I=\frac{-1}{2 \sqrt{3}} \tan ^{-1}\left(\frac{2 \cos x}{\sqrt{3}}\right)+c \\ & \end{aligned}$