Evaluate : \[\int_{0}^{\pi / 2} \frac{x}{\sin x+\cos x} d x \] Case-Study Based Question

\[\begin{aligned} I & =\int_{0}^{\pi / 2} \frac{x}{\sin x+\cos x} d x \\ & =\int_{0}^{\pi / 2} \frac{\left(\dfrac{\pi}{2}-x\right)}{\sin \left(\dfrac{\pi}{2}-x\right)+\cos \left(\dfrac{\pi}{2}-x\right)} d x \end{aligned} \] \[\begin{aligned} I & =\frac{\pi}{4 \sqrt{2}} \log \left|\sec \left(x-\frac{\pi}{4}\right)+\tan \left(x-\frac{\pi}{2}\right)\right|_{0}^{\frac{\pi}{2}} \\ & =\frac{\pi}{4 \sqrt{2}}[\log (\sqrt{2}+1)-\log (\sqrt{2}-1)] \end{aligned} \] Or \(\displaystyle I=\frac{\pi}{4 \sqrt{2}} \log \left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)\)