Integral of (t-b)^(x-1)(a-t)^(y-1)/(t-x)^(x+y) dt=(a-b)^(x+y-1)/((a-c)^x(b-c)^y) B(x,y)

Theorem

The following formula holds for $\mathrm{Re}(x)>0$, $\mathrm{Re}(y) > 0$, and $c<b<a$: $$\displaystyle\int_a^b \dfrac{(t-b)^{x-1}(a-t)^{y-1}}{(t-c)^{x+y}} \mathrm{d}t=\dfrac{(a-b)^{x+y-1}}{(a-c)^x (b-c)^y} B(x,y),$$ where $B$ denotes the beta function.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (14)$