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[parent] integration of differential binomial (Theorem)

Theorem. Let $ a$, $ b$, $ c$, $ \alpha$, $ \beta$ be given real numbers and $ \alpha\beta \neq 0$. The antiderivative

$\displaystyle I = \int x^a(\alpha+\beta x^b)^c\,dx$
is expressible by means of the elementary functions only in the three cases: $ (1)\,\, \frac{a+1}{b}+c\in\mathbb{Z}$, $ (2)\,\, \frac{a+1}{b}\in\mathbb{Z}$, $ (3)\,\, c\in\mathbb{Z}$

In accordance with P. L. Chebyshev (1821$ -$1894), who has proven this theorem, the expression $ x^a(\alpha+\beta x^b)^c\,dx$ is called a differential binomial.

It may be worth noting that the differential binomial may be expressed in terms of the incomplete beta function and the hypergeometric function. Define $ y = \beta x^b / \alpha$. Then we have

$\displaystyle I = {1 \over b} \alpha^{{a + 1 \over b} + c} \beta^{-{a + 1 \over b}} B_y \left( {1 + a \over b}, c - 1 \right)$
$\displaystyle = {1 \over 1 + a} \alpha^{{a + 1 \over b} + c} \beta^{-{a + 1 \ov... ...y^{1 + a \over b} F \left( {a + 1 \over b}, 2-c; {1 + a + b \over b}; y \right)$

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



"integration of differential binomial" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Cross-references: hypergeometric function, beta function, incomplete, terms, binomial, expression, elementary functions, expressible, antiderivative, real numbers
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This is version 2 of integration of differential binomial, born on 2004-10-23, modified 2004-10-25.
Object id is 6408, canonical name is IntegrationOfDifferentialBinomial.
Accessed 2261 times total.

Classification:
AMS MSC: 26A36 (Real functions :: Functions of one variable :: Antidifferentiation)
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