Question

$\int \frac{{\sin }^{2}x{\cos }^{2}x}{{({\sin }^{3}x+{\cos }^{3}x)}^2}$

Answer 1

$I=\int \frac{{\sin }^{2}x{\cos }^{2}x}{{\left({\sin }^{3}x+{\cos }^{3}x\right)}^2}dx=\int \frac{{\tan }^{2}x{\sec }^{2}x}{{\left(1+{\tan }^{2}x\right)}^2}dxPut1+{\tan }^{3}x=t\frac{dt}{dx}=3{\tan }^{2}x{\sec }^{2}xNowI=\int \frac{dt}{t^2}=\frac{-1}{3}{t}^{-1}+C=\frac{1}{3\left(1+{\tan }^{2}x\right)}+C.$