How Do You Evaluate the Limit of an Integral with a Root Function?

[dirk_mec1]

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?

 

[arildno]

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 $e^{\frac{n}{e}}$, occurring at $x=\frac{1}{e}$.
Thus, an upper bound for the integral is $e^{\frac{n}{e}}$, and as an upper bound for the whole expression we have:
$$\sqrt[n]{e^{\frac{n}{e}}}=e^{\frac{1}{e}$$

Thus, our limit lies somewhere between 1 and $e^{\frac{1}{e}}$

 

[ice109]

may i ask where this integral arises?

 

[nicksauce]

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

 

[snipez90]

Can the integral be evaluated through parametric integration or other multivariate calculus methods?

 

[dirk_mec1]

Can the integral be evaluated through parametric integration or other multivariate calculus methods?

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.