Monthly Archives: November 2025

In this post, we will prove the following monstrous looking identity from Gradshteyn and Ryzhik (3.255): $$ \int _0^1 \frac{x^{\mu+\frac{1}{2}} (1-x)^{\mu-\frac{1}{2}}}{(c+2bx-ax^2)^{\mu+1}}dx = \frac{\sqrt{\pi}}{\left\{a + \left(\sqrt{c+2b-a} + \sqrt{c}\right)^2\right\}^{\mu+\frac{1}{2}}\sqrt{c+2b-a}} \frac{\Gamma \left(\mu + \frac{1}{2}\right)}{\Gamma\left(\mu+1\right)}$$ where $c+2b-a>0$, $a + \left(\sqrt{c+2b-a} + \sqrt{c}\right)^2 > 0$ … Continue reading →