Evaluate : \[\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{2 / 3}} d x \]

\[\begin{aligned} I & =\int_{0}^{\pi / 2} \frac{d x}{1+(\tan x)^{2 / 3}} \\ & =\int_{0}^{\pi / 2} \frac{\cos ^{2 / 3} x}{\cos ^{2 / 3} x+\sin ^{2 / 3} x} d x \\ & =\int_{0}^{\pi / 2} \frac{\cos ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)}{\cos ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)+\sin ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)} d x \\ & =\int_{0}^{\pi / 2} \frac{\sin ^{2 / 3} x}{\sin ^{2 / 3} x+\cos ^{2 / 3} x} d x \\ & (\text { I })+(\text { II }) \\ & \Rightarrow 2 I=\int_{0}^{\pi / 2} 1 d x=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4} \end{aligned} \]