In some of my research - notably that relating to statistical distribution theory, and that in Bayesian econometrics - I spend quite a bit of time dealing with integration problems. As I noted in this recent post, integration is something that we really can't avoid in econometrics - even if it's effectively just "lurking behind the scenes", and not right in our face.
Contrary to what you might think, this can be rather interesting!
We can use software, such as Maple, or Mathematica, to help us to evaluate many complicated integrals. Of course, that wasn't always so, and in any case it's a pity to let your computer have all the fun when you could get in there and get your hands dirty with some hands-on work. Is there anything more thrilling than "cracking" a nasty looking integral?
I rely a lot on the classic book, Table of Integrals, Series and Products, by Gradshteyn Ryzhik. It provides a systematic tabulation of thousands of integrals and other functions. I know that there are zillions of books that discuss various standard methods (and non-standard tricks) to help us evaluate integrals. I'm not qualified to judge which ones are the best, but here's one that caught my attention some time back and which I've enjoyed delving into in recent months.
It's written by an electrical engineer, Paul J. Nahin, and it's called Inside Interesting Integrals.
I just love Paul's style, and I think that you will too. For instance, he describes his book in the following way -
"A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics. (Plus 60 Challenge Problems with Complete, Detailed Solutions.)"
Well, that certainly got my attention!
And then there's the book's "dedication":
"This book is dedicated to all who, when they read the following line from John le Carre´’s 1989 Cold War spy novel The Russia House, immediately know they have encountered a most interesting character:
'Even when he didn’t follow what he was looking at, he could relish a good page of mathematics all day long.'
as well as to all who understand how frustrating is the lament in Anthony Zee’s book Quantum Field Theory in a Nutshell:
'Ah, if we could only do the integral … . But we can’t.' "
What's not to love about that?
Take a look at Inside Interesting Integrals - it's a gem.