Binomial integral

A binomial integral is an integral of the form:

${\displaystyle \int x^{m}\left(ax^{n}+b\right)^{p}\,\mathrm {d} x}$ , where are rational numbers and . ${\displaystyle m,\ n,\ p}$ ${\displaystyle a\neq 0,n\neq 0}$

The set of Chebyshev now makes a statement when a binomial integral is elementary integrated. Elementary integrable means that the integral can be determined with the help of an antiderivative.

Chebyshev theorem

statement

A binomial integral can be elementarily integrated if and only if at least one of the numbers is or is whole . ${\displaystyle \textstyle p,\ {\frac {m+1}{n}}}$ ${\displaystyle \textstyle {\frac {m+1}{n}}+p}$

If the function can be integrated elementarily, the antiderivative can be determined in the following three cases :

1. ${\displaystyle p\in \mathbb {Z} }$with the substitution where q is the main denominator of m and n ${\displaystyle x=t^{q}}$
2. ${\displaystyle {\frac {m+1}{n}}\in \mathbb {Z} }$with the substitution where q is the denominator of p ${\displaystyle t^{q}=ax^{n}+b}$
3. ${\displaystyle {\frac {m+1}{n}}+p\in \mathbb {Z} }$with the substitution where q is the denominator of p. ${\displaystyle t^{q}={\frac {ax^{n}+b}{x^{n}}}}$

Examples

1st example

${\displaystyle {\begin{aligned}&\int {\frac {1}{\sqrt {1+x^{4}}}}\,\mathrm {d} x=\int x^{0}\left(1\cdot x^{4}+1\right)^{-{\frac {1}{2}}}\,\mathrm {d} x\\\Rightarrow &m=0,\ n=4,\ p=-{\frac {1}{2}}\\\Rightarrow &p=-{\frac {1}{2}}\notin \mathbb {Z} ,\ {\frac {m+1}{n}}={\frac {0+1}{4}}={\frac {1}{4}}\notin \mathbb {Z} ,\ {\frac {m+1}{n}}+p={\frac {0+1}{4}}-{\frac {1}{2}}=-{\frac {1}{4}}\notin \mathbb {Z} \end{aligned}}}$

It is therefore not fundamentally integrable. ${\displaystyle \textstyle {\frac {1}{\sqrt {1+x^{4}}}}}$

2nd example

${\displaystyle {\begin{aligned}&\int {\sqrt[{3}]{x}}{\sqrt[{3}]{\left(x+1\right)^{2}}}\,\mathrm {d} x=\int x^{\frac {1}{3}}\left(1\cdot x^{1}+1\right)^{\frac {2}{3}}\,\mathrm {d} x\\\Rightarrow &m={\frac {1}{3}},\ n=1,\ p={\frac {2}{3}}\\\Rightarrow &p={\frac {2}{3}}\notin \mathbb {Z} ,\ {\frac {m+1}{n}}={\frac {{\frac {1}{3}}+1}{1}}={\frac {4}{3}}\notin \mathbb {Z} ,\ {\frac {m+1}{n}}+p={\frac {{\frac {1}{3}}+1}{1}}+{\frac {2}{3}}=2\in \mathbb {Z} \end{aligned}}}$

So is elementary integrable. ${\displaystyle \textstyle {\sqrt[{3}]{x}}{\sqrt[{3}]{\left(x+1\right)^{2}}}}$

source

• binomial integral . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .