Express in terms of Euler integrals

Question

Express in terms of Euler integrals: $$ \int_{0}^{+\infty}\frac{x^{m-1}}{(1+x)^n} dx $$

Answer 1

Let $ t=\frac{1}{1+x} $

\begin{align} I&=\int_0^1 t^{n-m-1}(1-t)^{m-1} \,dt \\ &=B(n-m,m) \\ &=\frac{\Gamma(n-m) \Gamma(m)}{\Gamma(n)} \end{align}