Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of a integral

  1. Aug 8, 2008 #1
    How can you evaluate this integral?

    \lim_{n\to\infty}\sqrt[n]{\int_0^1 x^{-nx}\ dx}

    Are there any rules for taking a square root into an integral?
  2. jcsd
  3. Aug 8, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    No, that is not possible. If you take a root of a sum, can you say that this equals the sum of the roots of the terms? That's basically the same that you're asking!

    The above limit seems very nasty, but we can at least roughly estimate a bounding interval containing that limit value:

    Note that, for fixed n, the MINIMUM value of the integrand is 1. Therefore, a lower bound for the integral equals 1, and the n'th root of 1, i.e, 1, is a lower bound for the whole expression.

    Now, you may readily show that the maximum value for the integrand equals [itex]e^{\frac{n}{e}}[/itex], occurring at [itex]x=\frac{1}{e}[/itex].
    Thus, an upper bound for the integral is [itex]e^{\frac{n}{e}}[/itex], and as an upper bound for the whole expression we have:

    Thus, our limit lies somewhere between 1 and [itex]e^{\frac{1}{e}}[/itex]
    Last edited: Aug 8, 2008
  4. Aug 8, 2008 #3
    may i ask where this integral arises?
  5. Aug 8, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    Here is a graph I made in maple... looks like the limit is roughly 1.44.

    Attached Files:

  6. Aug 8, 2008 #5
    Can the integral be evaluated through parametric integration or other multivariate calculus methods?
  7. Aug 8, 2008 #6
    I don't know but in contrast to arildno's answer the exact solution can be via integration methods. I just don't which and how to apply them.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Limit of a integral
  1. Limit of integral (Replies: 3)

  2. Limits of integrals (Replies: 9)

  3. Limit of an integral (Replies: 12)

  4. Limits and Integrals (Replies: 2)

  5. Integration limits (Replies: 3)