In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quadratic integral is an integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 of the form


It can be evaluated by completing the square

Completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the formax^2 + bx + c\,\!to the formIn this context, "constant" means not depending on x. The expression inside the parenthesis is of the form ...

 in the denominator.

Positive-discriminant case

Assume that the discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

 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

This (hastily written) section may need attention.

In case the discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

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


is positive. Then the integral becomes