Integration of a real powered rational expression

Question

Peace be upon you,

I've encountered this pretty integral \begin{align*} \int_0^1&\frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}dx,\\ \\ &\alpha,\beta\in\Re^+ \end{align*} It seems much simpler than a question; but as I implied some of the known integration techniques like: Substitution, Polynomial division, Term by term integration, it was not solved.

Even I used Matlab Mupad and set the necessary assumptions and then performed the integration, but no results occurred.

Can anyone light up any idea?

Answer 1

You may write

$$\frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{1-x} = (x^{\alpha+\beta-1}-x^{\alpha-1})\sum_{k=0}^\infty x^k,$$

then

$$\begin{align} \int_0^1 \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{1-x}\mathrm{d}x &= \sum_{k=0}^\infty \int_0^1 (x^{\alpha+\beta-1}-x^{\alpha-1})x^k\mathrm{d}x\\\\ &= \sum_{k=0}^\infty \left(\frac{1}{k+\alpha+\beta} - \frac{1}{k+\alpha}\right)\\\\ &= - \psi(\alpha+\beta)+\psi(\alpha) \end{align}$$

by the well-known series representation of the $\psi$ function DLMF: here 5.7.6.

Answer 2

Using the same approach as Olivier Oloa in his answer :

• for the antiderivative $$\int \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}\,dx=x^{\alpha } \left(\frac{1}{\alpha }-\frac{x^{\beta }}{\alpha +\beta }\right)-B_x(\alpha +\beta +1,0)+B_x(\alpha +1,0)$$

• for the integral $$\int_0^1 \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}\,dx=\psi ^{(0)}(\alpha +\beta )-\psi ^{(0)}(\alpha )$$

Answer 3

Hint: The n-th harmonic number is $~H_n=\displaystyle\int_0^1\frac{1-x^n}{1-x~~}dx.~$ Some particular values can be found here.