# Difference between revisions of "Differential binomial"

From Encyclopedia of Mathematics

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An expression of the type | An expression of the type | ||

− | $$x^m(a+bx^n)^ | + | $$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, | where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial, | ||

− | $$\int x^m(a+bx^n)^ | + | $$\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). | 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). |

## Latest revision as of 20:23, 1 January 2019

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