Need a hint proving that integral converges

[Tricore]

I am trying to prove that \int_0^{\pi/4} (\tan x)^{-\alpha} dx is finite for 0 < \alpha < 1, where the integral denotes the Lebesgue integral with the Lebesgue measure. I've decided wether it converges for all other values of \alpha\in\mathbb{R}, but am simply stuck with this one.

I've tried using monotone convergence to integrate over the closed interval from 1/n to pi/4, where I can use the Riemann integral, but I can't figure out how to find an anti-derivative for the function. I've also tried finding a larger, integrable function but without luck.

Any hints would be much appreciated.

 

[micromass]

What happens if \alpha=1. Is that function integrable?

 

[Hurkyl]

The only difficulty in this problem is the poles that the integrand has at 0 and pi/4, right? So find another function that has the same poles, and study that function, and the difference between it and your integrand.