\(\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x}+\sqrt[3]{\operatorname{cosec} x}} d x=\)

Let
\(I =\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}}{\sqrt[3]{\sec x}+\sqrt[3]{\operatorname{cosec} x}} \mathrm{dx} ...(1) \)
\( =\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec \left(\frac{\pi}{2}-x\right)} \sqrt{\sec \left(\frac{\pi}{2}-x\right)+\sqrt[3]{\operatorname{cosec}\left(\frac{\pi}{2}-x\right)}}}{\sqrt[3]{\frac{\pi}{2}}} d x \)
\( \therefore \text { I }= \int_{0}^{\frac{\pi}{2}} \frac{\sqrt[3]{\sec x}+\sqrt[3]{\operatorname{cosec} x}} d x ...(2)\)
Adding equation (1) \& (2), we get
\(2 I=\int_{0}^{\frac{\pi}{2}} 1 d x=[x]_{0}^{\frac{\pi}{2}} \Rightarrow 2 I=\frac{\pi}{2}-0 \Rightarrow I=\frac{\pi}{4}\)