Let $I=\int_{0}^{1}\left[x\left(1-x^{2}\right)\right]^{1 / 3} d x$

(a) Express $I$ in terms of an integral of the form $\oint\left(z^{3}-z\right)^{1 / 3} d z$, where the path of integration is a large circle. You should explain carefully which branch of $\left(z^{3}-z\right)^{1 / 3}$ you choose, by using polar co-ordinates with respect to the branch points. Hence show that $I=\frac{1}{6} \pi \operatorname{cosec} \frac{1}{3} \pi$.

(b) Give an alternative method of evaluating $I$, by making a suitable change of variable and expressing $I$ in terms of a beta function.