Is this integral incomplete gamma function?

Question

I'm trying to solve the following integral: $$\int_{0}^{x} t^{s-1}e^t dt$$ where, $0 \leq x \leq 1, 0 \leq t \leq 1, s >0$. Although I know the following integral is an incomplete gamma function $$\gamma(s, x) = \int_{0}^{x} t^{s-1}e^{-t} dt$$ Can someone please explain me if we can solve the other integral?

Answer 1

What do you mean by "solve"?

The first integral diverges as $x \to \infty$ and the second converges.

Both can be computed by putting in the power series for $e^t$ and integrating term by term. The first integral behaves better computationally since all the terms are positive so there is no cancellation.

You can get increasingly accurate estimates of the first integral by repeated integration by parts, just as for the second.

Answer 2

Make $t=-u$ $$I=\int_{0}^{x} t^{s-1}e^t dt=-\int_{0}^{-x}e^{-u} (-u)^{s-1}\,du=(-1)^{-s} (\Gamma (s)-\Gamma (s,-x))$$