A family of integrals analytically solvable

Abstract

We consider the following family of definite integrals:

Its explicit computation cannot be found in the traditional table books for general values of the parameters. It cannot be computed by algebraic manipulators either. Mathematica 7.0 computes the integral for some particular (low) values of m. In this paper, we compute this family of integrals for general values of the parameters in terms of well-known special functions: gamma and exponential integrals. From this computation, it is then straightforward to design a code for its evaluation with Mathematica and incorporate those integrals into the Mathematica library of integrals. We have compared the speed of computation of that integral for m=1, 3 by using our code on the one hand and the command Integrate of Mathematica 7.0 on the other hand, concluding that our code is much faster.

Keywords:

• evaluation of definite integrals
• gamma function and exponential integral
• Mathematica codes

2010 AMS Subject Classifications :

• 33F05
• 41A30
• 33B2
• 44A15
• 68N99

Acknowledgements

The Ministerio de Educación y Ciencia (REF. MTM2010-21037) is acknowledged by its financial support.