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

Theorem

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

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (13)$