Differential binomial

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An expression of the type


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).


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