Quadratic integral explained

In mathematics, a quadratic integral is an integral of the form $\int \frac.$

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

$\int \frac = \frac \int \frac.$

Positive-discriminant case

Assume that the discriminant q = b² − 4ac is positive. In that case, define u and A by $u = x + \frac,$ and $-A^2 = \frac - \frac = \frac(4ac - b^2).$

The quadratic integral can now be written as $\int \frac = \frac \int \frac = \frac \int \frac.$

The partial fraction decomposition $\frac = \frac\!\left(\frac - \frac \right)$ allows us to evaluate the integral: $\frac \int \frac = \frac \ln \left(\frac \right) + \text.$

The final result for the original integral, under the assumption that q > 0, is $\int \frac = \frac \ln \left(\frac \right) + \text.$

Negative-discriminant case

In case the discriminant q = b² − 4ac is negative, the second term in the denominator in $\int \frac = \frac \int \frac.$ is positive. Then the integral becomes $\begin \frac \int \frac & = \frac \int \frac \\[9pt]& = \frac \int \frac \\[9pt]& = \frac \arctan(w) + \mathrm \\[9pt]& = \frac \arctan\left(\frac\right) + \text \\[9pt]& = \frac \arctan \left(\frac\right) + \text \\[9pt]& = \frac \arctan\left(\frac\right) + \text.\end$

References

Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
Book: Izrail Solomonovich . Gradshteyn . Izrail Solomonovich Gradshteyn . Iosif Moiseevich . Ryzhik . Iosif Moiseevich Ryzhik . Yuri Veniaminovich . Geronimus . Yuri Veniaminovich Geronimus . Michail Yulyevich . Tseytlin . Michail Yulyevich Tseytlin . Alan . Jeffrey . Daniel . Zwillinger . Victor Hugo . Moll . Victor Hugo Moll . Scripta Technica, Inc. . Table of Integrals, Series, and Products . . 2015 . October 2014 . 8 . English . 978-0-12-384933-5 . 2014010276 . 2016-02-21-->. Gradshteyn and Ryzhik.