# Math Help - integration problem

1. ## integration problem

can anybody help me with this integral ?

(attached as bitmap)

2. Hello, rbenito!

I got a different answer . . .

$\int^{\infty}_0 a\,x\,e^{-a}\,dx$

Integrate by parts . . .

. . $\begin{array}{ccccccc}u & = & x & \quad & dv & = & ae^{-ax}dx \\
du & = & dx & \quad & v & = & -e^{-ax}\end{array}$

We have: . $-xe^{-ax} + \int^{\infty}_0\!\! e^{-ax}dx \;\;=\;\;-xe^{-ax} - \frac{1}{a}e^{-ax}\,\bigg]^{\infty}_0 \;\;=\;\;-\frac{e^{-ax}}{a}(ax + 1)\,\bigg]^{\infty}_0$

. . $= \;\lim_{t\to\infty}\left[-\frac{ax+1}{ae^{ax}}\right]^t_0 \;= \;\lim_{t\to\infty}\left[-\frac{at+1}{ae^{at}} - \left(-\frac{1}{a}\right)\right] \;=\;0 - \left(-\frac{1}{a}\right) \;=\;\frac{1}{a}
$

3. Hey Soroban.

I believe you're right. I see my mistake.

After looking at it again, I get the same.