- #1
michonamona
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Integration by parts - Exponential distribution
Solve the following definite integral:
\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx
I'm asked to solve this integral. The solution is \lambda, although I'm not sure how this was done.
\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx
= \frac{1}{\lambda} \int^{\infty}_{0} x e^{-\frac{x}{\lambda}} dx
=\frac{1}{\lambda} \left( \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right), integration by parts.
The \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} term, by fundamental theorem of calculus is 0. Thus,
= - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right),
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 e^{-\frac{x}{\lambda}} , which, solving for 0 and infinity will yield -1.
Where have I gone wrong?
I appreciate your input.
M
Homework Statement
Solve the following definite integral:
\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx
I'm asked to solve this integral. The solution is \lambda, although I'm not sure how this was done.
Homework Equations
The Attempt at a Solution
\int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx
= \frac{1}{\lambda} \int^{\infty}_{0} x e^{-\frac{x}{\lambda}} dx
=\frac{1}{\lambda} \left( \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right), integration by parts.
The \left[ x e^{-\frac{x}{\lambda}} \right] ^{\infty}_{0} term, by fundamental theorem of calculus is 0. Thus,
= - \int^{\infty}_{0} e^{-\frac{x}{\lambda}} dx \right),
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 e^{-\frac{x}{\lambda}} , which, solving for 0 and infinity will yield -1.
Where have I gone wrong?
I appreciate your input.
M
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