Списак интеграла рационалних функција

Списак интеграла рационалних функција:

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

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

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

${\displaystyle \int {\frac {x\;dx}{ax+b}}={\frac {x}{a}}ax+b\right|}$

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

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

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

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

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

${\displaystyle \int {\frac {dx}{x(ax+b)}}={\frac {ax+b}{x}}\right|}$

${\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}={\frac {ax+b}{x}}\right|}$

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

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

${\displaystyle \int {\frac {dx}{x^{2a^{2}}}=x|<|a|{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {dx}{x^{2a^{2}}}=x|>|a|{\mbox{)}}\,\!}$

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

${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b\sqrt {b^{24ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{(for }}4ac-b^{2}<0{\mbox{)}}}$

${\displaystyle \int {\frac {x\;dx}{ax^{2}+bx+c}}={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|ax^{2}+bx+c\right|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{)}"}}' id="mwQQ">$${\displaystyle \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{)}}}$

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

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

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

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

Литература

title= (помоћ). 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. Недостаје или је празан параметар |title= (помоћ)

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)