# Probability and Laplace/Fourier transforms to solve limits/integrals from calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those answers, see also this for an example that has to do with evaluation of limit of a series) or some uses Laplace transforms or even Fourier series (Example see some answers by user Tunk-Fey: 1, 2 ). I would like to ask about a book that discusses those techniques, i.e. where to learn them?

$\bullet$ Edit: There's no problem if those techniques can only be learned from a particular section of one book or more (or even just a paper, as long as it is accessible to an upper-undergrad.).

Thanks in advance, that would really help me unlock my box of tools.

• The Laplace transform and Fourier series will be discussed in pretty much any differential equations textbook. – Chantry Cargill Dec 24 '14 at 20:54
• I guess "Inside Interesting Integrals: A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, " and Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals will be a good place to start searching – SomeOne Dec 24 '14 at 20:55
• @Ahmed I have both books, but none contains a treatment of probabilistic methods. However Nahin's book contains a small discussion of Fourier series and how they can be used to determine the value of an integral using one example. – user153330 Dec 24 '14 at 21:08
• @ChantryCargill My focus is not on the theory behind Laplace and Fourier transforms but on how they can be used to find integrals. – user153330 Dec 24 '14 at 21:13
• You might also check out the artofproblemsolving forum. I'm sure they have tons of arguments using the methods you've mentioned. – Robert Cardona Dec 24 '14 at 22:08