# Evaluating $\int_a^{\infty} x e^{-(x-a)} dx$

I cannot integrate $\int_a^{\infty} x e^{-(x-a)} dx$. I know the answer should be $(a+1)$ but when I use integration by parts I do not get that answer. Note that $a$ is a constant.

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Great. So now what? –  akkkk Apr 28 '12 at 15:06
By the way, the answer is not $(a+1)$. –  Did Apr 28 '12 at 15:10
Now that the post is modified, the answer is $(a+1)$. –  Did Apr 28 '12 at 15:29

$$\int x e^{-(x-a)} = e^a\int xe^{-x}$$
$$\int xe^{-x} = -xe^{-x} - \int -e^{-x} +c= -e^{-x}(x+1) +c$$ (applying Integration by parts)
so the final answer is, $$-e^{a-x}(x+1) +c.e^{a}$$