# Evaluate the Integral $$\int_{x_{0}}^{1} (1-x)^{a} x^{-a} dx$$

This problem was posed at Mathematics Stack Exchange, here is my solution.

\begin{align}
\int\limits_{x_{0}}^{1} (1-x)^{a} x^{-a} dx
&= \int\limits_{0}^{1} (1-x)^{a} x^{-a} dx – \int\limits_{0}^{x_{0}} (1-x)^{a} x^{-a} dx \\
&= \mathrm{B}(1-a,1+a) – \mathrm{B}_{x_{0}}(1-a,1+a) \\
&= \Gamma(1-a)\Gamma(1+a) – \frac{x_{0}^{1-a}}{1-a} \, {}_{2}\mathrm{F}_{1}(1-a,-1;2-a;x_{0})
\end{align}

We have used the incomplete beta function and Gauss’s hypergeometric function.

## 2 thoughts on “Evaluate the Integral $$\int_{x_{0}}^{1} (1-x)^{a} x^{-a} dx$$”

1. If you started by the substitution x -> 1-x the final answer will be simpler.

2. You are correct. The solution can be written as a single incomplete beta function or hypergeometric function.