# Список на интеграли на ирационални функции

Список на интеграли на ирационални функции:

## Интеграли со a² - x²

${\displaystyle \int {\sqrt {a^{2}-x^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {a^{2}-x^{2}}}+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$
${\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\;dx=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$
${\displaystyle \int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x}}={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$
${\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

## Интеграли со x² + a²

${\displaystyle \int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {x^{2}+a^{2}}}+a^{2}\,\mathrm {arcsinh} {\frac {x}{a}}\right)}$
${\displaystyle \int x{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}}$
${\displaystyle \int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x}}={\sqrt {x^{2}+a^{2}}}-a\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}$
${\displaystyle \int {\frac {dx}{\sqrt {x^{2}+a^{2}}}}=\mathrm {arcsinh} {\frac {x}{a}}=\ln \left|x+{\sqrt {x^{2}+a^{2}}}\right|}$
${\displaystyle \int {\frac {x\,dx}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}}$
${\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {x^{2}+a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\mathrm {arcsinh} {\frac {x}{a}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}+a^{2}}}\right|}$
${\displaystyle \int {\frac {dx}{x{\sqrt {x^{2}+a^{2}}}}}=-{\frac {1}{a}}\,\mathrm {arcsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}$

## Интеграли со x² - a²

${\displaystyle \int {\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}-\operatorname {sgn} x\,\mathrm {arccosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}$
${\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}$
${\displaystyle \int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x}}={\sqrt {x^{2}-a^{2}}}-a\arccos {\frac {a}{x}}\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\mathrm {arccosh} {\frac {x}{a}}=\ln \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}$
${\displaystyle \int {\frac {x\;dx}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}$
${\displaystyle \int {\frac {x^{2}\,dx}{\sqrt {x^{2}-a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\,\mathrm {arccosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}+a^{2}\ln \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\right)\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}$

## Интеграли со ax² + bx + c

${\displaystyle \int {{\sqrt {ax^{2}+bx+c}}\;dx}={\frac {1}{8a^{3/2}}}\left(2{\sqrt {a}}\cdot \left(b+2ax\right){\sqrt {c+x\left(b+ax\right)}}-\left(b^{2}-4ac\right)\log {\left(b+2x+2{\sqrt {a}}{\sqrt {c+x\left(b+ax\right)}}\right)}\right)}$
${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a(ax^{2}+bx+c)}}+2ax+b\right|\qquad {\mbox{(za }}a>0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\mathrm {arcsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(za }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\qquad {\mbox{(za }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(za }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}$
${\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}}$

## Литература

• Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
• I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)
• Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
• Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)