- #1
RyanSchw
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Improper Integral [Solved]
[tex]
\int_{0}^{\infty} (x-1)e^{-x}dx
[/tex]
Integration by Parts
Improper Integrals
[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
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
Last edited: