Mathematics - Indefinite Integration Question with Solution | TestHub

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

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

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

$=\int \frac{{\sin }^{2}x{\cos }^{2}x}{{\left({\cos }^{3}x\right)}^2{\left({\tan }^{3}x+1\right)}^2}dx$

$=\int \frac{{\tan }^{2}x{\sec }^{2}x}{{\left({\tan }^{3}x+1\right)}^2}dx$
Put,${\tan }^{3}x+1=t$
$\Rightarrow dt=3{\tan }^{2}x{\sec }^{2}xdx$
$=\int \frac{dt}{3{\left(t\right)}^2}$$=\frac{-1}{3t}+C$, where$C$is the constant of integration.
$= \frac{-1}{3\left({\tan }^{3}x+1\right)}+C$