Splitting an integral with an absolute value

[Koshi]

This is a physics problem, but I only need help with the calculus portion of it. I was having trouble figuring out how to split the integral to properly integrate.

Homework Statement

Homework Equations

\int\stackrel{\infty}{-\infty}(x/x_0)e^-2|x|/x_0dx

where x_o is a constant

The Attempt at a Solution

I was wondering how to write the lower part of the integral. What I have is

\int\stackrel{0}{-\infty}(x/x_0)e^2x/x_0dx+\int\stackrel{\infty}{0}(x/x_0)e^-2x/x_0dx

Is that right or should I keep the negative in front of the 2 in the lower integral?

 

[Dick]

You've got it right. But there is an easy way to do your integral. If f(x) is the integrand, how does f(x) compare with f(-x)?

 

[Mark44]

Fixed the LaTeX in your first integral. Double-click it to see what I did.
Here are some tips:
Use only a single pair of [ tex] and [ /tex] tags (without the leading spaces that I show) for a given expression. You had tex tags around almost every item.
Don't mix [ sup] and [ sub] tags inside [ tex] tags. They don't work. Instead use ^{} for superscripts and _{} for subscripts.

This is a physics problem, but I only need help with the calculus portion of it. I was having trouble figuring out how to split the integral to properly integrate.

Homework Statement

Homework Equations

\int_{-\infty}^{\infty}(x/x_0)e^{-2|x|/x_0} dx

where x_o is a constant

The Attempt at a Solution

I was wondering how to write the lower part of the integral. What I have is

\int\stackrel{0}{-\infty}(x/x_0)e^2x/x_0dx+\int\stackrel{\infty}{0}(x/x_0)e^-2x/x_0dx

Is that right or should I keep the negative in front of the 2 in the lower integral?

 

[Koshi]

Fixed the LaTeX in your first integral. Double-click it to see what I did.

Thanks for the fix. I was struggling with it :P

 

[Koshi]

You've got it right. But there is an easy way to do your integral. If f(x) is the integrand, how does f(x) compare with f(-x)?

I was thinking I could just double the integral from 0 to \infty but because there was an x/x_0 in front and I would have to integrate by parts, I reasoned against it.

I wouldn't know how else to simplify it.

 

[Dick]

I was thinking I could just double the integral from 0 to \infty but because there was an x/x_0 in front and I would have to integrate by parts, I reasoned against it.

I wouldn't know how else to simplify it.

Yes, if you want to do the integral from 0 to infinity you should integrate by parts. But you can't double it to get the whole integral because the integral from -infinity to 0 isn't the same. How are the two integrals related? You could, of course, just do the integration and find out... But you would be going to a lot of extra work.