How can i solve $\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau$,

Question

I want to find the value of the integral
$$\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau,$$

where $0<\alpha<1$. Using Mathematica I found the solution to be

$$\frac{\sqrt{\pi}}{2}\frac{\Gamma(1-\alpha)}{\Gamma(\frac{5}{2}-\alpha)}t^{\frac{3-2\alpha}{2}}.$$

Question: Could anyone please help me solve this problem or give me some hints? Please, I have no idea as to how exactly I can find the value of the integral.

Best..

Answer 1

Hint. After the change of variable $$u=\frac{\tau}{t}$$ use the Euler beta integral: $$ B(a,b)=\int_{0}^{1}(1-u)^{a-1} u^{b-1}\,du = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}. $$