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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

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begin{align}
int_{0}^{infty}{x^{a - 1} - x^{b - 1} over 1 - x},dd x & =
int_{0}^{1}{x^{a - 1} - x^{b - 1} over 1 - x},dd x +
int_{1}^{infty}{x^{a - 1} - x^{b - 1} over 1 - x},dd x
end{align}
In the RHS second integral I'll perform the change $ds{x mapsto 1/x}$:
begin{align}
&bbox[10px,#ffd]{int_{0}^{infty}{x^{a - 1} - x^{b - 1} over 1 - x},dd x}
\[5mm] = &
int_{0}^{1}{x^{a - 1} - x^{b - 1} over 1 - x},dd x +
int_{1}^{0}{pars{1/x}^{a - 1} - pars{1/x}^{1 - b} over 1 - 1/x}
,{dd x over -x^{2}}
\[5mm] = &
int_{0}^{1}{x^{a - 1} - x^{b - 1} over 1 - x},dd x -
int_{0}^{1}{x^{-a} - x^{-b} over 1 - x},dd x
\[5mm] & =
bbx{ds{H_{b - 1} - H_{a - 1} + H_{-a} - H_{-b}}}
quadmbox{with} Repars{a},, Repars{b} in pars{0,1}
end{align}
$ds{H_{n}}$ is a Harmonic Number and I used a well known identity ( as given by Euler ): $ds{H_{z} = int_{0}^{1}{1 - t^{z} over 1 - t},dd t}$ with $ds{Repars{z} > -1}$.

For $0<a,b<1$, with $x=e^u$
begin{eqnarray}
int_{0}^{infty}frac{x^{a-1}-x^{b-1}}{1-x}dx
&=&
int_{-infty}^{infty}frac{e^{au}-e^{bu}}{1-e^u}dx\
&=&
int_{0}^{infty}frac{e^{au}-e^{bu}}{1-e^u}dx+int_{0}^{infty}frac{e^{(1-a)u}-e^{(1-b)u}}{1-e^u}dx\
&=&
int_{0}^{infty}frac{-e^{-u}}{1-e^{-u}}Big(e^{au}-e^{bu}+e^{(1-a)u}-e^{(1-b)u}Big)dx\
&=&
int_{0}^{infty}sum_{n=1}^infty e^{-nu}Big(e^{au}-e^{bu}+e^{(1-a)u}-e^{(1-b)u}Big)dx\
&=&
sum_{n=1}^inftyint_{0}^{infty}Big(e^{(a-n)u}-e^{(b-n)u}+e^{(1-a-n)u}-e^{(1-b-n)u}Big)dx\
&=&
sum_{n=1}^inftyBig(frac{1}{n-a}-frac{1}{n-b}+frac{1}{n+a-1}-frac{1}{n+b-1}Big)
end{eqnarray}

You have to study what happens near $x=1$ and when $xto infty$, as I suppose you have notice.

Defining $f(x)=frac{x^{a-1}-x^{b-1}}{1-x}$, $lim_{xto 1} f(x)$ 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.