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:

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).

( Z stands for J or Y; in case a = n π, the sum is zero );

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`