# 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
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## 1 Answer

$$\int x e^{-(x-a)} = e^a\int xe^{-x}$$

$$\int xe^{-x} = -xe^{-x} - \int -e^{-x} +c= -e^{-x}(x+1) +c$$ (applying Integration by parts)

so the final answer is, $$-e^{a-x}(x+1) +c.e^{a}$$

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Confusing a number and a function. –  Did Apr 28 '12 at 15:20
I changed the function to x-a in the power term –  lord12 Apr 28 '12 at 15:23
Evaluating from a to infinity the answer is a+1. –  lord12 Apr 28 '12 at 15:27
lord12: Then you can copy the answer above, with this modification in mind, and see what happens. –  Did Apr 28 '12 at 15:28
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