Integrals

This page is a collection of various interesting integration results. While it currently focuses on what might be called the algebraic theory, namely of “doing” definite integrals and finding primitives, future expansion is planned to include some actual Analysis.

G.H. Hardy, in a letter of November 1926 to D. Coxeter, detailing the evaluation of some integrals the latter requested in the Mathematical Gazette:

I tried very hard not to spend time on your integrals, but to me the challenge of a definite integral is irresistible.

Said integrals were^[1]

Shameless self-promotion:

• \( \displaystyle \int_{0}^{\infty } (1+x) \arctan x \, (\log x)^{4} \frac{dx}{\sqrt{x}(1+x^2)} = \frac{57\pi^6\sqrt{2}}{64} \) : MSE
• \( \displaystyle \int_{0}^{\infty} \frac{\arctan x \, \log(1+x^2)}{x(1+x^2)} \, dx = \frac{\pi}{2} \log^{2} 2 \): MSE
• \( \displaystyle \int_0^{\infty} \frac{\sqrt{e^x-1}}{2\cosh{x}-1}\, dx = { \frac{\pi^2}{6}+ \frac{\pi}{2 \sqrt{3}} \log\big(7+4 \sqrt{3}\big) } \) : MSE
• \( \displaystyle \int_{0}^{\pi/2} \log\left( \frac{a+b\sin^2 x}{a-b\sin^2 x }\right) \frac{dx}{\sin^2 x} = \left( \sqrt{1+\frac{b}{a}} - \sqrt{1-\frac{b}{a}} \right)\pi \) : MSE
• \( \displaystyle \int_{-\infty}^\infty \frac{\log\big(\sqrt{x^2+a^2}\big)}{x^2+b^2} \, dx = \frac{\pi}{b}\log(a+b) \) : MSE
• \( \displaystyle \int \frac{x \sin x \cos x}{(x-\sin x \cos x)^2} \, dx = \frac{ \cos^2 x }{ 2( \sin x \cos x - x ) } + C \) : MSE
• \( \displaystyle \int_{-\infty}^{+\infty}\sin(\cosh x)\cos(\sinh x)\mathrm dx = \frac{\pi}{2} \) : MSE
• \( \displaystyle \int \frac{ e^{\arctan x }}{(1+x^2)^{3/2}} \, dx = \frac{1+x}{2\sqrt{1+x^2}}e^{\arctan{x}} + C \) : MSE
• \( \displaystyle \int_{-\pi}^\pi(2+2\cos t)^a\cos bt \, dt = \frac{2\pi\Gamma(1+2a)}{\Gamma(1+a+b)\Gamma(1+a-b)} \) : MSE

A short list of some obscure integration gadgetry

• The Cauchy formula for repeated integration
• Ramanujan’s master theorem
• Glasser’s master theorem
• The Gudermannian function
• Differentiation under the integral (known to many from Surely you’re joking, Mr Feynman)

Useful books and links

• Digital Library of Mathematical Functions: repository of special function information. Includes a decent selection of integrals, although by no means exhaustive.
• Gradshteyn and Ryzhik’s Table of Integrals, Series and Products. The table of integrals. So extensive, it’s genuinely difficult to find what you want in it.
• I’m not generally a fan of the various books that have sprung up recently about evaluating integrals. Some are good, but some are retreads of the same basic material, a lot of which any motivated calculus student will see. An incomplete list, of ones I have looked at myself:
  • Irresistable Integrals by Boros and Moll. This is quite a good one, although it does lean a bit heavily on using Mathematica. Moll has written a lot of papers about verifying the integrals in Gradshteyn and Ryzhik, see his website and his arXiv page.
  • Inside Interesting Integrals by Nahin. I don’t like this one. The typesetting is terrible, for one thing, and the author does not cite his sources. He also has what feels like a lack of faith in the mathematics: if you have proven it, why do you need to check it on an unreliable computer?
  • How to Integrate It: A Practical Guide to Finding Elementary Integrals by Stewart. Clearly designed with a complete beginner in mind. Does not go far beyond A-level techniques.
  • The Integration of Functions of a Single Variable by G.H. Hardy. For many years a reasonably definitive text on finding primitives (AKA indefinite integrals). There are better resources available now, although I don't have any to hand.
• Ron Gordan’s integration blog A master of the Residue Theorem practises his art.
• A more extensive list is available in this Mathematics Stack Exchange answer.

• How to integrate \( \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx\)? A cautionary lesson in indefinite integration: even for a relatively simple irrational function, the algorithm is a complete nightmare.

Some of these may be an interesting source of a summer project, should a student be interested.

• Ramanujan’s master theorem and the Method of Brackets: The Method of Brackets is a rather odd formal integration technique invented/discovered quite recently, with applications in evaluating the notoriously difficult integrals in Quandum Field Theory that arise from Feynman diagrams, as well as more generally. Is there an actual proof of the algorithm employed by the Method, which apperas to be somewhat arbitrary? There is a set of relavent articles available on arXiv.
• How much useful integration theory can be done with the upper and lower Darboux integrals? For example, are there any generalisations of the convergence theorems for the Riemann or Lebesgue integral? One such is obtained by Bullen and Výborný,^[2] but it seems there is more to do here.
• What is the Kolmogorov integral?

Some Mathematics Stack Exchange questions on integration without a satisfactory answer:

• Is the integral form of the polarisation identity useful for anything?
• Can this proof of the second MVT for integrals be salvaged?