Difference between revisions of "Differential binomial"
From Encyclopedia of Mathematics
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An expression of the type | An expression of the type | ||
| − | + | $$x^m(a+bx^n)^pdx,$$ | |
| − | where | + | where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial, |
| − | + | $$\int x^m(a+bx^n)^pdx,$$ | |
| − | is reduced to an integral of rational functions if at least one of the numbers | + | is reduced to an integral of rational functions if at least one of the numbers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integral of a differential binomial cannot be expressed by elementary functions (P.L. Chebyshev, 1853). |
Revision as of 16:33, 4 November 2014
An expression of the type
$$x^m(a+bx^n)^pdx,$$
where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,
$$\int x^m(a+bx^n)^pdx,$$
is reduced to an integral of rational functions if at least one of the numbers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integral of a differential binomial cannot be expressed by elementary functions (P.L. Chebyshev, 1853).
Comments
The statement on the reduction to an integral of rational functions is called the Chebyshev theorem on the integration of binomial differentials.
How to Cite This Entry:
Differential binomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_binomial&oldid=11396
Differential binomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_binomial&oldid=11396
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article