Faraday effect

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

$\int \frac{dx}{a+bx+c{x}^2}=\frac{1}{\sqrt{q}}\ln \left(\frac{2cx+b-\sqrt{q}}{2cx+b+\sqrt{q}}\right)+\text{constant, where }q=b^2-4ac.$

Negative-discriminant case

This (hastily written) section may need attention.

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:
• Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.