Help integrating $\int_0^t\lambda e^{(a+b-1)\lambda x}(1-e^{-\lambda x})^{b-1}dx$

Question

How to calculate the integral $$\int_0^t\lambda e^{(a+b-1)\lambda x}(1-e^{-\lambda x})^{b-1}dx~,$$ where all $\lambda,\;a$ and $b$ are constants? WolframAlpha won't solve it, but is there some kind of simple trick for this?

Answer 1

Force $u=1-\exp -\lambda x$ so your integral is an incomplete Beta function, viz. $$\int_0^{1-\exp -\lambda t}u^{b-1}(1-u)^{-a-b}du=B(1-\exp -\lambda t;\,b,\,-a-b+1).$$