Integration with exponential functions for hydrogen energy

[ronaldoshaky]

Homework Statement

$$E = \frac{e^2}{4 \pi \epsilon_0 a_0^3 } \left( - b \int_0^b e^{- \frac{2r}{a_0}} dr + \int_0^b r e^{- \frac{2r}{a_0}} dr \right)$$

I have to use these integrals,

$$\int_0^x e^{-u} du = 1 - e^{-x}$$ and $$\int_0^x u e^{-u} du = 1 - e^{-x} - xe^{-x}$$

I get:

The Attempt at a Solution

$$E = \frac{e^2}{4 \pi \epsilon_0 a_0^3 } \left( - \frac{a_0 b}{2} (1 - e^{- \frac{2b}{a_0}}) + \left(- \frac{- a_0 b}{2} e^{- \frac{2b}{a_0}} - \frac{a_0^2}{4} e^{- \frac{2b}{a_0}} + \frac{a_0^2}{4} \right) \right)$$

 

[phyzguy]

I think you blew a sign in the third term. It should be negative, not positive. Otherwise, it looks correct.