Question

Evaluate $\int \frac{{\cos }^{2}x}{{\sin }^{3}x{\left({\sin }^{5}x+{\cos }^{5}x\right)}^{\frac{3}{5}}}dx$

Answer 1

$I=\int \frac{{\cos }^{4}x}{{\sin }^{3}x{\left({\sin }^{5}x+{\cos }^{5}x\right)}^{\frac{3}{5}}}dx$

$I=\int \frac{{\sec }^{2}xdx}{{\tan }^{6}x{\left(1+\frac{1}{{\tan }^{5}x}\right)}^{\frac{3}{5}}}$

Let

$\tan x=p{\sec }^{2}xdx=dp$

Therefore,

$\int \frac{dp}{p^6{\left(1+\frac{1}{p^5}\right)}^{\frac{3}{5}}}1+\frac{1}{p^5}=k\frac{dk}{dp}=\frac{-5}{p^6}$

Therefore,

$I=-\frac{1}{5}\int {\left(k\right)}^{\frac{-3}{5}}dk=\frac{-1}{5}\left(\frac{5}{2}\right)k^{\frac{2}{5}}+C=\frac{-1}{2}{\left[\frac{p^5+1}{p^5}\right]}^{\frac{2}{5}}=-\frac{1}{2}{\left[\frac{{\tan }^{5}x+1}{{\tan }^{5}x}\right]}^{\frac{2}{5}}+C=-\frac{1}{2}{\left(1+{\cot }^{5}x\right)}^{\frac{2}{5}}+C$

Hence, this is the answer.