# integration of differential binomial

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

$$I=\int {x}^{a}{(\alpha +\beta {x}^{b})}^{c}\mathit{d}x$$ |

is expressible by 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

$$I=\frac{1}{b}{\alpha}^{\frac{a+1}{b}+c}{\beta}^{-\frac{a+1}{b}}{B}_{y}(\frac{1+a}{b},c-1)$$ |

$$=\frac{1}{1+a}{\alpha}^{\frac{a+1}{b}+c}{\beta}^{-\frac{a+1}{b}}{y}^{\frac{1+a}{b}}F(\frac{a+1}{b},2-c;\frac{1+a+b}{b};y)$$ |

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

Title | integration of differential binomial |
---|---|

Canonical name | IntegrationOfDifferentialBinomial |

Date of creation | 2013-03-22 14:45:49 |

Last modified on | 2013-03-22 14:45:49 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 26A36 |