Integrate e^-a|x|: Troubleshooting & Solution

[dotman]

Homework Statement

I'm a little confused with this integral:

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

Homework Equations

The Attempt at a Solution

Now, I believe the typical way to evaluate this is to say, hey, because of the |x|, this thing is symmetric about the x axis, and so we can instead evaluate:

$$\int^\infty_{-\infty}e^{-a|x|}\,dx = 2\int^\infty_0e^{-ax}\,dx$$
$$= 2[-\dfrac{1}{a}e^{-ax}|^\infty_0] \, \, = \, 2[\dfrac{1}{a}] \, \, = \, \dfrac{2}{a}$$

which I believe is correct. However, and this is my question, can it be evaluated without using this trick? I ran into trouble, and I'm not sure where I made my mistake, although I suspect it has to do with not really doing anything about the absolute value of x:

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

$$= \, \, [-\dfrac{1}{a}e^{-a|x|}|^\infty_{-\infty}]$$

$$= \, \, -\dfrac{1}{a}[e^{-a|\infty|} - e^{-a|-\infty|}]$$

$$= \, \, -\dfrac{1}{a}[e^{-a\infty} - e^{-a\infty}]$$

$$= \, \, -\dfrac{1}{a}[0 - 0] \, = \, 0$$

And I've gotten nowhere, but I can't tell why, or what mistake I committed (if any).

Thoughts? What am I missing here? Thanks!

 

[lanedance]

the first way is correct it comes form the fact
$$e^{-a|x|} = e^{-ax},x\geq 0$$
$$e^{-a|x|} = e^{ax},x < 0$$

so the integral becomes
$$\int^\infty_{-\infty}e^{-a|x|}dx = \int^{0}_{-\infty}e^{ax}dx + \int^{\infty}_{0}e^{-ax}dx$$

which simplifies to what you gave (use substitution u = -x in first part)

your 2nd interegral is not valid due to the different behaivour of $e^{-a|x|}$ either side of zero