\(\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=I\) माना .....(i)
\(I=\frac{\sqrt{\cos \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)} t \sqrt{\cos \left(\frac{\pi}{2}-x\right)}}=\frac{\sqrt{\sin x}}{\sqrt{\cos x}+\sqrt{\sin x}} \) ......(ii)
समीकरण (i) और (ii) को जोड़ने पर,
\(2 I =\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}+\frac{1}{\sqrt{\cos }}\)
\(=\int_{0}^{\frac{\pi}{2}} d x=[x]_{0}^{\frac{\pi}{2}}=\left[\frac{\pi}{2}-x\right]=\frac{\pi}{2}\)
\(2 I=\frac{\pi}{2} ; \)
\(I=\frac{\pi}{4}\)
