integration of differential binomial

Theorem.  Let a, b, c, α, β be given real numbers and  αβ0.  The antiderivative


is expressible by of the elementary functionsMathworldPlanetmath only in the three cases:   (1)a+1b+c,   (2)a+1b,   (3)c

In accordance with P. L. Chebyshev (1821-1894), who has proven this theorem, the expression  xa(α+βxb)cdx  is called a differential binomial.

It may be worth noting that the differential binomial may be expressed in terms of the incomplete beta functionDlmfDlmfDlmfMathworld and the hypergeometric functionDlmfDlmfDlmfMathworldPlanetmath. Define y=βxb/α. Then we have


Chebyshev’s theorem then follows from the theorem on elementary cases of the hypergeometric function.

Title integration of differential binomial
Canonical name IntegrationOfDifferentialBinomial
Date of creation 2013-03-22 14:45:49
Last modified on 2013-03-22 14:45:49
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Theorem
Classification msc 26A36