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.