Списак интеграла рационалних функција

Списак интеграла рационалних функција:

$\int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!$

$\int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|$

$\int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{(for }}n\not \in \{-1,-2\}{\mbox{)}}$

$\int {\frac {x\;dx}{ax+b}}={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|$

$\int {\frac {x\;dx}{(ax+b)^{2}}}={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|$

$\int {\frac {x\;dx}{(ax+b)^{n}}}={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{(for }}n\not \in \{-1,-2\}{\mbox{)}}$

$\int {\frac {x^{2}\;dx}{ax+b}}={\frac {1}{a^{3}}}\left({\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\ln \left|ax+b\right|\right)$

$\int {\frac {x^{2}\;dx}{(ax+b)^{2}}}={\frac {1}{a^{3}}}\left(ax+b-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)$

$\int {\frac {x^{2}\;dx}{(ax+b)^{3}}}={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)$

$\int {\frac {x^{2}\;dx}{(ax+b)^{n}}}={\frac {1}{a^{3}}}\left(-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(a+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad {\mbox{(for }}n\not \in \{1,2,3\}{\mbox{)}}$

$\int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|$

$\int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|$

$\int {\frac {dx}{x^{2}(ax+b)^{2}}}=-a\left({\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right)$

$\int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!$

$\int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {artanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\mbox{(for }}|x|<|a|{\mbox{)}}\,\!$

$\int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arcoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\mbox{(for }}|x|>|a|{\mbox{)}}\,\!$

$\int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}4ac-b^{2}>0{\mbox{)}}$

$\int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{(for }}4ac-b^{2}<0{\mbox{)}}$

$\int {\frac {x\;dx}{ax^{2}+bx+c}}={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}$

$\int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}4ac-b^{2}>0{\mbox{)}}$

$\int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}4ac-b^{2}<0{\mbox{)}}$

$\int {\frac {dx}{(ax^{2}+bx+c)^{n}}}={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac {(2n-3)2a}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!$

$\int {\frac {x\;dx}{(ax^{2}+bx+c)^{n}}}={\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!$

$\int {\frac {dx}{x(ax^{2}+bx+c)}}={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {dx}{ax^{2}+bx+c}}$

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, ур. (2007). Table of Integrals, Series, and Products. seventh edition. Academic Press. ISBN 978-0-12-373637-6. CS1 одржавање: Вишеструка имена: списак уредника (веза). 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 978-2-88124-097-3.  . 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 978-1-58488-956-4. 
• Zwillinger, Daniel (2002). CRC Standard Mathematical Tables and Formulae. 31st edition. Chapman & Hall/CRC Press. ISBN 978-1-58488-291-6. . (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)