_0^ /4 ^2x ^2x( ^3x+ ^3x)^2dx

Question

$\int _{0}^{\pi /4}\frac{{\cos }^{2}x{\sin }^{2}x}{{({\cos }^{3}x+{\sin }^{3}x)}^{2}}dx$

Accepted Answer

$\begin{gathered} \int _{0}^{\pi /4}\frac{{\cos }^{2}x\cdot {\sin }^{2}x}{{({\cos }^{3}x+{\sin }^{3}x)}^{2}}dx \ =\int _{0}^{\pi /4}\frac{{	an }^{2}x\cdot {\sec }^{2}x}{{(1+{	an }^{3}x)}^{2}}dx \end{gathered}$

Let $	an x=t$

$\int _{0}^{1}\frac{{t}^{2}dt}{{(1+{t}^{3})}^{2}}$

Let $1+{t}^{3}=z$

$\begin{gathered} 3{t}^{2}dt=dz \ \frac{1}{3}\int _{1}^{2}\frac{dz}{{z}^{2}}={\frac{1}{3}(-\frac{1}{z})|}_{2}^{1} \ =-\frac{1}{3}(\frac{1}{2}-1)=\frac{1}{6} \end{gathered}$

Solutions · 1