$ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt[3]{\sin{x}}}{\sqrt[3]{\sin{x}}+\sqrt[3]{\cos{x}}} dx$

Question

How to proceed after the below step. Also is there a simpler way to solve this integral?

score 3 · Accepted Answer · answered Jun 09 '16 at 13:37

Hint: The substitution $x=\frac{\pi}{2}-\theta$ gives a very similar integral over the same range. If you sum the original integral and the last one, you get that the original integral is $\frac{1}{2}\int_{\pi/6}^{\pi/3}1\,dx = \frac{\pi}{12}$. Symmetry trick.

$ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt[3]{\sin{x}}}{\sqrt[3]{\sin{x}}+\sqrt[3]{\cos{x}}} dx$

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