# Homework Help: Integrating absolute values over infinity

1. Nov 7, 2012

### ElijahRockers

1. The problem statement, all variables and given/known data

Find <x> in terms of X0 if X0 is constant and

$\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}$

and

$<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx$

where Psi* is the complex conjugate of Psi.

Since there is no imaginary component, this is effectively Psi2.

so, from here I could do a u-substitution to integrate over e^u du, but I'm not sure how.

What is the derivative of -2|x|/X_0 with respect to x?

This is part of a physics exercise i'm working on.

$<x> = \frac{1}{x_0}\int^{\infty}_{-\infty}e^{\frac{-2|x|}{X_0}} dx$

I have found that the derivative of |x| depends on whether x<0 or x>0. for x<0, x'=-1 and for x>0, x'=1 but I'm not sure how to tie this all together for the integration.

I guess what I'm really asking is how do I find the integrand here?

Last edited: Nov 7, 2012
2. Nov 7, 2012

### SammyS

Staff Emeritus
So, split the integral.

$\displaystyle <x> = \frac{1}{x_0}\int^{0}_{-\infty}xe^{\frac{-2|x|}{X_0}} dx+\frac{1}{x_0}\int^{\infty}_{0}xe^{\frac{-2|x|}{X_0}} dx$