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, again

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]

\[ \begin{align} \int_{0}^{\pi/2} \sec^{-1}(\sec x + 2) \, dx &= \frac{5\pi^2}{24} \\ \int_{0}^{\pi/3} \sec^{-1}(2\cos x + 1) \, dx &= \frac{\pi^2}{8} \\ \int_{0}^{\pi/3} \cos^{-1}(\sec x - 1) \, dx &= \frac{11\pi^2}{72} . \end{align} \]

Some nasty integrals I have evaluated

Shameless self-promotion:

Useful information

A short list of some obscure integration gadgetry

Useful books and links

Worth looking at

Some things I am thinking about

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

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