Expected r^2 for the 2s wavefunction of hydrogen atom

[chall]

Homework Statement

Calculate the expected value for r^2 for the 2s wavefunction of the hydrogen atom (only the radial part of the function is needed for l=0). If you choose to solve this problem graphically, plot or sketch the function you integrate.

Homework Equations

R(r)=1/sqrt(2a^3)*(1-r/2a)*e^(-r/2a) where a=bohr radius

The Attempt at a Solution

to calculate, i know you integrate (from 0 to inf) as follows: int((R(r))^2*r^2 dr),
but I'm having trouble solving the integral. I'm not sure how I would solve it graphically either.

Any help is appreciated. Thanks

 

[vela]

If you're familiar with the gamma function, you can use

$$\Gamma(n) = \int_0^\infty t^{n-1}e^{-t}\,dt = (n-1)!$$

If you need to show that result, you can integrate by parts to prove it by induction. Or you can use this trick:

$$\int_0^\infty t^ne^{-\alpha t}\,dt = \int_0^\infty \left(-\frac{d}{d\alpha}\right)^n e^{-\alpha t}\,dt = \left(-\frac{d}{d\alpha}\right)^n \int_0^\infty e^{-\alpha t}\,dt$$

Do the integral, differentiate, and then set $\alpha=1$.