Quadratic integral


In mathematics, a quadratic integral is an integral of the form

It can be evaluated by completing the square in the denominator.

Positive-discriminant case

Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by


The quadratic integral can now be written as

The partial fraction decomposition

allows us to evaluate the integral:

The final result for the original integral, under the assumption that q > 0, is

Negative-discriminant case

In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in

is positive. Then the integral becomes


  • Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
  • Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.