Quadratic Integral

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'' = ''b''² − 4''ac'' 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'' = ''b''² − 4''ac'' 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 \\ pt& = \frac \int \frac \\ pt& = \frac \arctan(w) + \mathrm \\ pt& = \frac \arctan\left(\frac\right) + \text \\ pt& = \frac \arctan \left(\frac\right) + \text \\ pt& = \frac \arctan\left(\frac\right) + \text.
\end

References

*Weisstein, Eric W.
Quadratic Integral
" From ''MathWorld''--A Wolfram Web Resource, wherein the following is referenced:
*

{{Calculus topics

Integral calculus