Integral Involving L'Hopital's Rule

Recently, I've been doing several practice problems on a website called [HTML]http://www.math.ucdavis.edu/~kouba/ProblemsList.html[/HTML] I haven't yet taken an official calculus course, but I've found many of the problems I've studied on this website so far to be quite simple. The problem that's troubling be is number 8, which can be viewed on [HTML]http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/defintdirectory/DefInt.html[/HTML] I've understood the linked solution (below the question) up until the point where it requires L'Hopital's Rule to be used. The problem is this:

From my understanding, L'Hopitals rule involves the constant differentiation of a function to a point where that function is no longer in an indeterminate form. Where do I even begin here? Do I start off by finding the derivative of ? Help would me much appreciated.