Question

Integrate $\int \frac{{\cos }^{4}x}{{\sin }^{3}x{({\sin }^{5}x+{\cos }^{5}x)}^{3/5}}dx$

Answer 1

$=\int \frac{{\cos }^{4}x.dx}{{\sin }^{6}x{[1+\frac{{\cos }^{5}x}{{\sin }^{5}x}]}^{\frac{3}{5}}}$

Multiplying Numerator and denominator by ${\cos }^{2}x$

$=\int \frac{{\cos }^{4}x.dx.{\cos }^{2}x}{{\cos }^{2}x.{\sin }^{6}x{[1+\frac{{\cos }^{5}x}{{\sin }^{5}x}]}^{\frac{3}{5}}}dx$

$=\int \frac{{\sec }^{2}x.dx}{{\tan }^{6}x{[1+\frac{1}{{\tan }^{5}x}]}^{\frac{3}{5}}}.dx$

Let $\tan x=P\Rightarrow {\sec }^{2}x.dx=dp$

$I=\int \frac{dp}{p^6{[1+\frac{1}{p^5}]}^{\frac{3}{5}}}$ $1+\frac{1}{p^5}=R$

$I=-\frac{1}{5}\int R^{-\frac{3}{5}}$ $-5\frac{1}{P^6}.dp=dR$

$=-\frac{1}{5}\times \frac{5}{2}{R}^{\frac{2}{5}}+C=-\frac{1}{2}{R}^{\frac{2}{5}}+C$

$=-\frac{1}{2}{\left[\frac{p^5+1}{p^5}\right]}^{\frac{2}{5}}+C$

$I=-\frac{1}{2}{\left[\frac{{\tan }^{5}x+1}{{\tan }^{5}x}\right]}^{\frac{2}{5}}+C$