# Improper integral involving sinc function and Pochhammer symbol

Can anyone please advise me how to integrate expressions of the form $\text{sinc}\,(x) / (1-x)_n$ along the real axis?

Using a CAS, one could suggest that $$n! \int_{-\infty}^\infty \frac{\sin \pi x}{\pi x (1-x)_n}dx=2^n,\ n=1,2,\ldots$$ But I don't know how to prove this.

• A few manipulations give the equivalent integral $$\int_{-\infty}^\infty \frac{\Gamma(1+n)}{\Gamma(1+t)\Gamma(1-t+n)}\mathrm dt=\int_{-\infty}^\infty \binom{n}{t}\mathrm dt$$ – J. M. is a poor mathematician Apr 10 '16 at 10:50
• Thank you for the interesting observation, but how about integrals of binomial coefficients? Could you please suggest some source on how to integrate them? – user2835965 Apr 10 '16 at 12:19