Evaluation of $\int_{0}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$ 
For $0<a,b<1.$
  Evaluation of $$\int_{0}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$$

$\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{x^{a-1}-x^{b-1}}{1-x}dx+\int_{1}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$$
Now how can i proceed further, Help required, Thanks

You have to study what happens near $x=1$ and when $x\to \infty$, as I suppose you have notice. 
Defining $f(x)=\frac{x^{a-1}-x^{b-1}}{1-x}$, $\lim_{x\to 1} f(x)$ exists and it's finite (why?). So you can define $f(1)$ by continuity, and $\int_0^1 f(x)\ dx$ exists and it's a number.
I do not know how to proceed from here. Any help would be appreciated.
