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.