Quadratic integral - Alchetron, The Free Social Encyclopedia

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

Positive-discriminant case

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

u = x + b 2 c ,

and

− A 2 = a c − b 2 4 c 2 = 1 4 c 2 ( 4 a c − b 2 ) .

The quadratic integral can now be written as

∫ d x a + b x + c x 2 = 1 c ∫ d u u 2 − A 2 = 1 c ∫ d u ( u + A ) ( u − A ) .

The partial fraction decomposition

1 ( u + A ) ( u − A ) = 1 2 A ( 1 u − A − 1 u + A )

allows us to evaluate the integral:

1 c ∫ d u ( u + A ) ( u − A ) = 1 2 A c ln ⁡ ( u − A u + A ) + constant .

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

∫ d x a + b x + c x 2 = 1 q ln ⁡ ( 2 c x + b − q 2 c x + b + q ) + constant, where q = b 2 − 4 a c .

Negative-discriminant case

This (hastily written) section may need attention.

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

∫ d x a + b x + c x 2 = 1 c ∫ d x ( x + b 2 c ) 2 + ( a c − b 2 4 c 2 ) .

is positive. Then the integral becomes

1 c ∫ d u u 2 + A 2 = 1 c A ∫ d u / A ( u / A ) 2 + 1 = 1 c A ∫ d w w 2 + 1 = 1 c A arctan ⁡ ( w ) + c o n s t a n t = 1 c A arctan ⁡ ( u A ) + constant = 1 c a c − b 2 4 c 2 arctan ⁡ ( x + b 2 c a c − b 2 4 c 2 ) + constant = 2 4 a c − b 2 arctan ⁡ ( 2 c x + b 4 a c − b 2 ) + constant .

References

Quadratic integral Wikipedia

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Contents

Positive-discriminant case

Negative-discriminant case

References