The value of $\int \sqrt[3]{\frac{\sin ^n x}{\cos ^{n+6} x}} d x$, is

$\frac{3}{n+3} \tan ^{n / 3} x+C$ $\frac{3}{n+3} \tan ^{n / 3+1} x+C$ $\frac{3}{n+1} \tan ^{n / 3+1} x+C$ none of these

$\frac{3}{n+3} \tan ^{n / 3+1} x+C$

We have, $I=\int \sqrt[3]{\frac{\sin ^n x}{\cos ^{n+6} x}} d x=\int 3 \sqrt{\tan ^n x} \sec ^2 x d x$ $\Rightarrow I=\int \tan ^{n / 3} x d(\tan x)=\frac{(\tan x)^{\frac{n}{3}+1}}{\frac{n}{3}+1}=\frac{3}{n+3} \tan ^{\frac{n}{3}+1} x+C$