Question

How do you evaluate the integral `int xe^-x dx` from 0 to `oo`?

Answer 1

1

Explanation:

`I (alpha) = int_0^oo e^(-alpha x) dx = [-1/alpha e^(- alpha x)]_0^oo= 1/alpha`

`(dI)/(d alpha) = int_0^oo -x e^(-alpha x) dx = -1/alpha^2`

`implies int_0^oo x e^(-alpha x) dx = 1/alpha^2`

`implies int_0^oo x e^(-x) dx = 1`