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I need INTegral from 0 to ∞ of : x*e^(-x) dx

I use IBP: u = x, du = dx

dv = e^(-x) dx, v = -e^(-x)

uv - ∫v du = -x*e^(-x) - ∫ -e^(-x) dx

I am trying to evaluate the uv term : -x*e^(-x) from 0 to ∞

for the ∞ term of -x*e^(-x) =

-inf * 1/e^(inf) = -inf * 1/inf = 1

OR

-inf * 1/e^(inf) = -inf * 1/inf = -inf * 0 = 0

OR

it is indeterminate form, use l hospital...

which path is correct one ?

for the 0 term of -x*e^(-x) = 0

I am really sorry for the noob calculus question.

This is part of larger question regarding Gauss Quadrature. but..

Kindly pls remind me of the correct one.

Thank you for your help

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# Noob Integral Question, Indeterminate form of ?

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