Question

Let I = $\int \frac{{\sin }^{2}x{\cos }^{2}x dx}{\left[\right({\sin }^{2}x+{\cos }^{2}x\left)\right({\sin }^{3}x+{\cos }^{3}x){]}^2}$
=$\int \frac{{\sin }^{2}x{\cos }^{2}x}{({\sin }^{3}x+{\cos }^{3}x){}^2}dx$
=$\int \frac{{\tan }^{2}x.se{c}^{2}x}{(1+{\tan }^{3}x{)}^3}dx$
put (1 + tan³ x) = t
3 tan² x sec²x dx = dt
$I=\frac{1}{3}\int \frac{dt}{t^2}=-\frac{1}{3t}+C$
Hence, I = $\frac{-1}{3(1+{\tan }^{3}x)}+C$