TITLE: Closed representation of this integral
QUESTION [1 upvotes]: I was wondering whether there is some easy closed representation for $\int_R^{\infty} e^{-k(r-R)}r^{l+1} dr$, where $l\in \mathbb{N_0}$ $k>0$ and $R>0$.

REPLY [1 votes]: $$ I=\int_R^{\infty} e^{-k(r-R)}r^{l+1} dr= e^{kR}\int_R^{\infty} e^{-kr}r^{l+1} dr. $$
Now, using the change of variables $kr=t$ gives
$$ = e^{kR}\int_R^{\infty} e^{-kr}r^{l+1} dr = \frac{e^{kR}}{k^{l+2}}\int_{Rk}^{\infty} e^{-t}t^{l+1} dr $$

$$I = \frac{e^{kR}}{k^{l+2}}\Gamma( l+2, Rk ),$$

where $ \Gamma(s,x) $ is the incomplete gamma function.
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