How do I normalize the wave equation?

[C. Darwin]

Homework Statement

\Psi(x) = \frac{C}{a^2 + x^2}

Homework Equations

I know to do this I need to solve for:
\int_{-\infty}^{\infty} \left|\Psi(x)\right|^2 = 1

The Attempt at a Solution

I'm not sure how to do it for this function. I've tried various methods to solve
C^2 \int_{-\infty}^{\infty} \frac{dx}{(a^2 + x^2)^2} = 1

 

[C. Darwin]

By substiting
x =a * tan(\theta)
I was able to solve for
\frac{C^2}{a^3} = \frac{2}{\pi}

I now need to find the expectation values of x and x^2 I'm not sure how to get started (are they just zero? if I plot this function I get a curve peaked at x=0...)

 

[Donaldos]

E[X]=\int\limits_{-\infty}^{+\infty} x \left|\Psi(x)\right|^2 {\rm d}x

f(x)=\Psi^2(x) is even and g(x)=x is odd \Rightarrow E[X]=\int\limits_{-\infty}^{+\infty} g(x)f(x) {\rm d}x= 0

E[X^2]=\int\limits_{-\infty}^{+\infty} x^2 \left|\Psi(x)\right|^2 {\rm d}x =?

 

[C. Darwin]

Thanks, I got it. Am I supposed to post a full solution?

 

[Donaldos]

I don't know but it would be nice.