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In mathematics, a quadratic integral is an integral of the form
![{\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89a9f8c6833bcfb3020c930f28a708915e4b5c1d)
It can be evaluated by completing the square in the denominator.
![{\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2cb1c59f47f9118759a906f5b2aef09ee60757)
Positive-discriminant case[edit]
Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by
![{\displaystyle u=x+{\frac {b}{2c}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b7a7e78a7aea9120a57f9dc0927fb86175de8e)
and
![{\displaystyle -A^{2}={\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}={\frac {1}{4c^{2}}}(4ac-b^{2}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba87cb2c9bacb3cf1a0bc917f7814e2b9c04e4c4)
The quadratic integral can now be written as
![{\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {du}{u^{2}-A^{2}}}={\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a91beea3886c87c263249032421631a83c7674a0)
The partial fraction decomposition
![{\displaystyle {\frac {1}{(u+A)(u-A)}}={\frac {1}{2A}}\!\left({\frac {1}{u-A}}-{\frac {1}{u+A}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2487d7bcea53f3b6630339575eb8a7ff450c167)
allows us to evaluate the integral:
![{\displaystyle {\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}={\frac {1}{2Ac}}\ln \left({\frac {u-A}{u+A}}\right)+{\text{constant}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5eeefe9d683ed6cabe8e86fab2ec6a3990b270)
The final result for the original integral, under the assumption that q > 0, is
![{\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{\sqrt {q}}}\ln \left({\frac {2cx+b-{\sqrt {q}}}{2cx+b+{\sqrt {q}}}}\right)+{\text{constant}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b874d28c0419bb977737e10258c93d5a3a21f7b3)
Negative-discriminant case[edit]
In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in
![{\displaystyle \int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2cb1c59f47f9118759a906f5b2aef09ee60757)
is positive. Then the integral becomes
![{\displaystyle {\begin{aligned}{\frac {1}{c}}\int {\frac {du}{u^{2}+A^{2}}}&={\frac {1}{cA}}\int {\frac {du/A}{(u/A)^{2}+1}}\\[9pt]&={\frac {1}{cA}}\int {\frac {dw}{w^{2}+1}}\\[9pt]&={\frac {1}{cA}}\arctan(w)+\mathrm {constant} \\[9pt]&={\frac {1}{cA}}\arctan \left({\frac {u}{A}}\right)+{\text{constant}}\\[9pt]&={\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}}\\[9pt]&={\frac {2}{\sqrt {4ac-b^{2}\,}}}\arctan \left({\frac {2cx+b}{\sqrt {4ac-b^{2}}}}\right)+{\text{constant}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0715d439a69d30944bda613d3112a8b890555207)
References[edit]
- Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.