Table of Integrals

A. Dieckmann, Physikalisches Institut der Uni Bonn

This integral table contains hundreds of expressions: indefinite and definite integrals of elliptic integrals, of square roots, arcustangents and a few more exotic functions. Most of them are not found in Gradsteyn-Ryzhik.

Sometimes m, n, k denote real parameters and are restricted mostly to 0 < {m, n, k} < 1, at times they represent natural numbers.

Results may be valid outside of the given region of parameters, but should always be checked numerically!

Definite Integrals:

Substitute      and the Feynman-Hibbs Integral can be calculated with Mathematica:

The expression below containing OwenT functions shows up as common  term

in the following six integrals :

(The integral above got cracked on my daughter' s birthday, so I call it Charlotte' s Integral .)

To see a nice cancellation of singularities at work plot the next expression around c = negative Integer:

…this is a special case of the next integral below (m = -1 / 2).

Arclength of of a helix around a torus with major radius R, minor radius r and number of windings n :

--  4 times the above integral gives the circumference of Cassini' s curve  --

in the following expressions (∫ f(x)/(a x^2 + b x + c ) dx)   we abbreviate s = :

the values at integer n can be found approximately by setting n near to an integer .

in the following expressions (∫ f(x)/(a x^4 + b x^2 + c ) dx)   we abbreviate s = :

Master formula of Boros and Moll:

Here the result is a threefold sum shown in Mathematica syntax:
KSubsets[aList, k] is in Package DiscreteMath`Combinatorica` and gives a list of all subsets with k elements of aList .
For n=3 the sum is .
<<DiscreteMath`Combinatorica`

use  Γ[1 - t] Γ[t] = π Csc[π t]