Tricky integral involving the gamma function

Question

Evaluate the integral $$\int_{-\infty}^k \frac{\Gamma\left(\frac{v+1}2\right)}{\sqrt{v\pi}\,\Gamma\left({\frac v2}\right)} \left(1+\frac{t^2}{v}\right)^{-\frac{v+1}2} dt$$

How would we start solving this integral? Would we use a trigonometric substitution?

Thanks.

Answer 1

This is the CDF at $k$ of a Student's-$t$ distribution. The answer involves a hypergeometric function. It can be obtained in terms of an incomplete Beta function with $t=\sqrt{v}\tan\theta$ followed by $u=\sin^2\theta$, i.e. $u =\frac{t^2}{v+t^2}$.