Quadratic integral

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

:$$int frac{dx}{a+bx+cx^2}.

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

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

Positive-discriminant case

Assume that the discriminant "q" = "b"² − 4"ac" 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} left( 4ac - b^2 ight).

The quadratic integral can now be written as

:$$int frac{dx}{a+bx+cx^2} = frac1c int frac{du}{u^2-A^2} = frac1c int frac{du}{(u+A)(u-A)}.

The partial fraction decomposition

:$$frac{1}{(u+A)(u-A)} = frac{1}{2A} left( frac{1}{u-A} - frac{1}{u+A} ight)

allows us to evaluate the integral:

:$$frac1c int frac{du}{(u+A)(u-A)} = frac{1}{2Ac} ln left( frac{u - A}{u + A} ight) + mathrm{constant}.

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

:$$int frac{dx}{a+bx+cx^2} = frac{1}{ sqrt{q ln left( frac{2cx + b - sqrt{q{2cx+b+ sqrt{q ight) + mathrm{constant},mbox{ where } q = b^2 - 4ac.

Negative-discriminant case

:"This (hastily written) section may need attention."

In case the discriminant "q" = "b"² − 4"ac" is negative, the second term in the denominator in

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

is positive. Then the integral becomes

::$$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{constant}

:$$ = frac{1}{cA} arctanleft(frac{u}{A} ight) + mathrm{constant}

:$$ = frac{1}{csqrt{frac{a}{c} - frac{b^2}{4c^2} arctanleft(frac{x + frac{b}{2c{sqrt{frac{a}{c} - frac{b^2}{4c^2} ight) + mathrm{constant}

:$$ = frac{2}{sqrt{4ac - b^2, arctanleft(frac{2cx + b}{sqrt{4ac - b^2 ight) + mathrm{constant}.

References

*Weisstein, Eric W. " [http://mathworld.wolfram.com/QuadraticIntegral.html 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.