grandes-ecoles 2021 Q17

grandes-ecoles · France · x-ens-maths2__mp Reduction Formulae Establish an Integral Identity or Representation

Let $\Gamma$ be the Gamma function. Show that for all $x \in ]0,1[$: $$\int_0^{+\infty} \frac{t^{x-1}}{1+t} dt = \frac{\pi}{\sin(\pi x)}.$$

Let $\Gamma$ be the Gamma function. Show that for all $x \in ]0,1[$:
$$\int_0^{+\infty} \frac{t^{x-1}}{1+t} dt = \frac{\pi}{\sin(\pi x)}.$$