Recently I've been trying to tackle the integral $\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$ using the Beta function
$$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\theta)d\theta=\frac{\sqrt{\pi}}{2}\left(\Gamma\left(\frac{x+1}{2}\right)\right)^{-1}$$
Differentiating both sides
$$\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))\sin^{x-1}(\theta)d\theta=-\frac{\sqrt{\pi}}{4}\frac{\psi(\frac{x+1}{2})}{\Gamma(\frac{x+1}{2})}$$
However, at $x=1$ $$\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta\ne\frac{\gamma\sqrt{\pi}}{4}$$
Where did I go wrong?
Best Answer
You forgot the $\Gamma(x/2)$ factor.