Question

Find the limit of the improper integral

Answer 1

\[\int\limits_{0}^{\infty}(x-1)e^ (-x)\]

Answer 2

Find antiderivative first

Answer 3

= xe^(-x) -e^(-x)

now, apply integration by parts to the first part

let u= x dv = e^(-x) dx

du= dx v= -e^(-x)


integral of first expression is uv - integral ( v du )

= -xe^(-x) + integral e^(-x) dx = -xe^(-x) -e^(-x)

Answer 4

that takes care of the first expression, but remember we still have to integrate the second expression.

integral of the second expression is e^(-x)

so sum them up

Answer 5

then sub in limits , you get zero im fairly sure