If I=_0^()/(2) ^(3)/(2) x ^(3)/(2) x+ ^(3)/(2) x ~d x, then _0^2I (x x x)/( ^4 x+ ^4 x) ~d…

Question

If $\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} \mathrm{~d} x$ equals :

Accepted Answer

For I

Apply king ($\mathrm{P}-5$) and add

$$\begin{aligned}
& 2 I=\int_0^{\pi / 2} d x=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4} \
& I_2=\int_0^{\pi / 2} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x
\end{aligned}$$

Apply king and add

$$\begin{aligned}
& \mathrm{I}_2=\frac{\pi}{4} \int_0^{\pi / 2} \frac{\tan \mathrm{xsec}{ }^2 \mathrm{xdx}}{\tan ^4 \mathrm{x}+1} \
& \text { put } \tan ^2 \mathrm{x}=\mathrm{t} \
& \frac{\pi}{8} \int_0^{\infty} \frac{\mathrm{dt}}{\mathrm{t}^2+1} \
& =\frac{\pi}{8} \cdot \frac{\pi}{2}=\frac{\pi^2}{16}
\end{aligned}$$

Solutions · 1