QJEEM25Ma8ae2: JEE Main 2025 Mathematics - Class 12 JEEM25Ma8ae2 Solution: If , then equals :

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

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