Quadratic integral

In mathematics, a quadratic integral is an integral of the form $$\int \frac{dx}{a+bx+c{x}^2}.$$

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

$$\int \frac{dx}{a+bx+c{x}^2}=\frac{1}{c}\int \frac{dx}{{(x+\frac{b}{2c})}^2+(\frac{a}{c}-\frac{b^2}{4c^2})}.$$

Positive-discriminant case

Assume that the discriminant q = b² − 4ac is positive. In that case, define u and A by $$u=x+\frac{b}{2c},$$ and $$-A^2=\frac{a}{c}-\frac{b^2}{4c^2}=\frac{1}{4c^2}(4ac-b^2).$$

The quadratic integral can now be written as $$\int \frac{dx}{a+bx+c{x}^2}=\frac{1}{c}\int \frac{du}{u^2-A^2}=\frac{1}{c}\int \frac{du}{(u+A)(u-A)}.$$

The partial fraction decomposition $$\frac{1}{(u+A)(u-A)}=\frac{1}{2A}(\frac{1}{u-A}-\frac{1}{u+A})$$ allows us to evaluate the integral: $$\frac{1}{c}\int \frac{du}{(u+A)(u-A)}=\frac{1}{2Ac}\ln \left(\frac{u-A}{u+A}\right)+\text{constant}.$$

The final result for the original integral, under the assumption that q > 0, is $$\int \frac{dx}{a+bx+c{x}^2}=\frac{1}{\sqrt{q}}\ln \left(\frac{2cx+b-\sqrt{q}}{2cx+b+\sqrt{q}}\right)+\text{constant}.$$

Negative-discriminant case

In case the discriminant q = b² − 4ac is negative, the second term in the denominator in $$\int \frac{dx}{a+bx+c{x}^2}=\frac{1}{c}\int \frac{dx}{{(x+\frac{b}{2c})}^2+(\frac{a}{c}-\frac{b^2}{4c^2})}.$$ is positive. Then the integral becomes $$\begin{array}{rl}\frac{1}{c}\int \frac{du}{u^2+A^2}& =\frac{1}{cA}\int \frac{du/A}{(u/A{)}^2+1}\\ & =\frac{1}{cA}\int \frac{dw}{w^2+1}\\ & =\frac{1}{cA}\arctan (w)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\\ & =\frac{1}{cA}\arctan \left(\frac{u}{A}\right)+\text{constant}\\ & =\frac{1}{c\sqrt{\frac{a}{c}-\frac{b^2}{4c^2}}}\arctan \left(\frac{x+\frac{b}{2c}}{\sqrt{\frac{a}{c}-\frac{b^2}{4c^2}}}\right)+\text{constant}\\ & =\frac{2}{\sqrt{4ac-b^2}}\arctan \left(\frac{2cx+b}{\sqrt{4ac-b^2}}\right)+\text{constant}.\end{array}$$

References

• Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
• (in English) Table of Integrals, Series, and Products (8 ed.). Academic Press, Inc.. 2015. ISBN 978-0-12-384933-5.