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1. The problem statement, all variables and given/known data

Solve the following definite integral:

[tex]\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx[/tex]

I'm asked to solve this integral. The solution is [tex]\lambda[/tex], although I'm not sure how this was done.

2. Relevant equations

3. The attempt at a solution

[tex]\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx[/tex]

[tex]= \frac{1}{\lambda} \int^{\infty}_{0} x e^{-\frac{x}{\lambda}} dx[/tex]

[tex]=\frac{1}{\lambda} \left( \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right) [/tex], integration by parts.

The [tex] \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} [/tex] term, by fundamental theorem of calculus is 0. Thus,

[tex]= - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right) [/tex],

I don't know what to do at this point, because as far as I know, taking the definite integral of this term will result in [tex]e^{-\frac{x}{\lambda}}[/tex] , which, solving for 0 and infinity will yield -1.

Where have I gone wrong?

I appreciate your input.

M

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# Homework Help: Integration by parts

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