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

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Answer 1 · 2016-09-24T11:19:43

1

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

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

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

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

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