Solving Integral for <x> with Ψ(x) = Ce^(-a(abs(x-2)))

[S_Flaherty]

I have a wave function Ψ(x) = Ce^(-a(abs(x-2))) and I have to find <x>.

I know that <x> = ∫x|ψ|2 dx (where the integral is from -Inf to Inf) so
<x> = ∫xCe^(-2a(abs(x-2)))dx where this C is actually C^2 but I can just make
it a new constant C and bring it out of the integral so I have C∫xe^(-2a(abs(x-2)))dx.

I can't think of anyway to solve the integral so either I solved |ψ|2 incorrectly or there is
some shortcut for solving an integral like that that I just don't know of.

Can someone tell me whether or not I'm actually doing the correct steps so far? And if so,
is there some kind of substitution I can use for solving the integral?

 

[TSny]

Try a substitution by letting u be something that will allow you to write the exponential function as $e^{-|u|}$.

 

[S_Flaherty]

Try a substitution by letting u be something that will allow you to write the exponential function as $e^{-|u|}$.

If I make -2a|x-2| equal to -|2ax-4a| I can substitute u = 2ax-4a so du = 2a dx, which leads to 1/2a du = dx. That makes the exponential function $e^{-|u|}$ but the integral is now
∫x$e^{-|u|}$du so I don't know how you can solve this since there is an x left which is not a constant.

 

[TSny]

Use u = 2ax-4a to express x in terms of u.

 

[S_Flaherty]

Use u = 2ax-4a to express x in terms of u.

Okay so I did that, solved the integrals and got <x> = 0, do you know/think this is correct?

 

[TSny]

No, you shouldn't get 0. You can tell what the answer should be by graphing the wavefunction (or the square of the wavefunction).

What did the integral(s) look like after making the substitution for u?