Definite Integral of Exponential Function

[singular]

[SOLVED] Definite Integral of Exponential Function

Homework Statement

I have an integral that I need to solve for a quantum physics problem

$$\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx$$

How would I go about solving this thing?

 

[Rainbow Child]

Split it into two intervals, i.e. $(-\infty,0),\,(0,\infty)$ and make a change or variables in the first one $x\to-x$

 

[singular]

Split it into two intervals, i.e. $(-\infty,0),\,(0,\infty)$ and make a change or variables in the first one $x\to-x$

$$\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx$$

Split into two intervals
$$\int^{\infty}_{0}e^{-a|x| - ikx}dx + \int^{0}_{-\infty}e^{-a|x| - ikx}dx$$

Change of variables in the second term x to -x
$$\int^{\infty}_{0}e^{-a|x| - ikx}dx - \int^{0}_{\infty}e^{-a|x| + ikx}dx$$

$$\int^{\infty}_{0}e^{-a|x| - ikx}dx + \int^{\infty}_{0}e^{-a|x| + ikx}dx$$

Are these steps what you are talking about?
What would I do from here?

 

[Rainbow Child]

Since $x\in(0,\infty)\Rightarrow |x|=x$. Now combine the two integrals.

 

[singular]

Since $x\in(0,\infty)\Rightarrow |x|=x$. Now combine the two integrals.

Oh...duh...thank you