Quadratic integral

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

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

Positive-discriminant case[edit]

Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by

,

and

The quadratic integral can now be written as

The partial fraction decomposition

allows us to evaluate the integral:

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

Negative-discriminant case[edit]

This (hastily written) section may need attention.

In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in

is positive. Then the integral becomes

References[edit]