If I= _( )/(2)^0 ^(3)/(2)x ^(3)/(2)x+ ^(3)/(2)xdx, then _2I^0x x x ^4x+ ^4xdx equals :

Question

If $I={\int }_{\frac{\pi }{2}}^{0}\frac{{\sin }^{\frac{3}{2}}x}{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}dx$, then ${\int }_{2I}^{0}\frac{x\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx$ equals :

Accepted Answer

For I

Apply king ($P-5$) and add

$\begin{gathered} 2I={\int }_{\pi /2}^{0}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4} \ {I}_{2}={\int }_{\pi /2}^{0}\frac{x\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx \end{gathered}$

Apply king and add

$\begin{gathered} {I}_{2}=\frac{\pi }{4}{\int }_{\pi /2}^{0}\frac{	an xsec{}^{2}xdx}{{	an }^{4}x+1} \ {	an }^{2}x=t \ \frac{\pi }{8}{\int }_{\infty}^{0}\frac{dt}{{t}^{2}+1} \ =\frac{\pi }{8}\cdot \frac{\pi }{2}=\frac{{\pi }^{2}}{16} \end{gathered}$

Solutions · 1