How to find $\int {t^n \, e^{t}}\mathrm dt$?

Question

Consider:$$\int {t^n e^{t}}\ \mathrm dt$$ is there any closed formula for this? W|A gave me this but I don't know what is Gamma function: $$\int {t^n e^t\ \mathrm dt} = (-t)^{-n}\ t^n\ \Gamma(n+1, -t)+ \text{constant}$$

Answer 1

Hint: Assuming $n$ is a positive integer, there is a closed formula including a sum $$\int t^n e^t \mathrm{d}t = n!e^t\sum_{k=0}^n (-1)^{n-k}\frac{t^k}{k!} + C $$ I got it by using the formulas (1) and (2) for the incomplete Gamma function. Once you have it, it can be verified by taking the derivative or you can try proving it with mathematical induction.

Answer 2

It is easily proved by induction and integration by parts that $$ \int {t^n e^{t}}\,dt = p_n(t) e^t $$ where $p_n$ is a polynomial of degree $n$ and $$ p_{n+1}(t) = t^{n+1}-(n+1)p_n(t) $$ This will give you the formula mentioned by gammatester.