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Integrals By Parts With Infinity As Limit

  1. Sep 29, 2010 #1
    [tex] \int_0^\infty \lambda x e^{-\lambda x} dx[/tex]

    How do I use the limits (infinity and 0) after getting the equation from integration by parts?
  2. jcsd
  3. Sep 29, 2010 #2
    Just do the limits. Remember, if lambda > 0, polynomials dominate at x = 0, and exponentials dominate at infinity.
  4. Sep 29, 2010 #3


    Staff: Mentor

    You need to write your improper integral as the limit of a proper integral.
    [tex] \int_0^\infty \lambda x e^{-\lambda x} dx = \lim_{b \to \infty} \int_0^b \lambda x e^{-\lambda x} dx [/tex]

    After you get your antiderivative, evaluate it at b and 0, and take the limit as b --> infinity.
  5. Oct 5, 2010 #4
    No need to evaluate it. The integral is the first moment (mean) of an exponential distribution on x, so it is equal to [tex]\lambda^{-1}[/tex]
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