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**Improper Integral [Solved]**

## Homework Statement

[tex]

\int_{0}^{\infty} (x-1)e^{-x}dx

[/tex]

## Homework Equations

Integration by Parts

Improper Integrals

## The Attempt at a Solution

[tex]

\lim_{R\rightarrow \infty} \int_0^R~xe^{-x}-e^{-x}dx

[/tex]

Let u = x

du = dx

Let dv = e^-x

v = -e^-x

[tex]

-xe^{-x} - \int -e^{-x}dx

[/tex]

[tex]

-xe^{-x}-e^{-x} - \int e^{-x}dx

[/tex]

[tex]

= -xe^{-x}

[/tex]

[tex]

\lim_{R\rightarrow \infty}-xe^{-x} \mid_{0}^{R}=0

[/tex]

Now there is clearly area when I look at the graph, but the graph intersects the x-axis at x=1 not x=0 I’m wondering if this is some sort of trick with the bounds of integration that I don’t see or if I made a mistake with my integration.

Thanks

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