# Differential binomial

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An expression of the type

$$x^m(a+bx^n)^p\,dx,$$

where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,

$$\int x^m(a+bx^n)^p\,dx,$$

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=43637
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article