From Wikipedia, the free encyclopedia
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
and
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.