how to integrate a function like this: $int_0^x frac{(t)^{a-1}}{(x-t)^a} dt$

Question

I have tried to do it myself and then looked for any hint here, but I can't reach the solution. Is it possible that it simplifies to a sine function when integrated between the limits 0 and x.
$$int_0^xfrac{(t)^{a-1}}{(x-t)^a} dt$$
where $0<a<1$

Angina Seng · Answer

Let $x>0$. An obvious substitition gives
$$int_0^xfrac{t^{a-1}}{(x-t)^a}, dt=int_0^1 u^{a-1}(1-u)^{-a},du.$$
This is a beta integral and equals
$$B(a,1-a)=frac{Gamma(a)Gamma(1-a)}{Gamma(1)}=fracpi{sinpi a}.$$

how to integrate a function like this: $int_0^x frac{(t)^{a-1}}{(x-t)^a} dt$

One Answer

Add your own answers!

Ask a Question