Struggling with the integrability of $\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$ I know quite a lot tools to determine the integrability of functions, but in this case I really don't know where to start:
$$\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$$

HINT:
$$\tan x=\frac{\sin x}{\cos x}=O\left((x-\pi/2)^{-1}\right)$$
and the integral
$$\int_0^1 \frac{dx}{x^a}$$
converges for $a<1$.
