Lista integrala racionalnih funkcija

Slijedi popis integrala (antiderivacija funkcija) racionalnih funkcija. Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala.

${\displaystyle \int (ax+b)^{n}dx}$	${\displaystyle ={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{(za }}n\neq -1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {1}{ax+b}}dx}$	${\displaystyle ={\frac {1}{a}}\ln \left|ax+b\right|}$
${\displaystyle \int x(ax+b)^{n}dx}$	${\displaystyle ={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{(za }}n\not \in \{-1,-2\}{\mbox{)}}}$

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

${\displaystyle \int {\frac {x^{2}}{ax+b}}dx}$	${\displaystyle ={\frac {1}{a^{3}}}\left({\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\ln \left|ax+b\right|\right)}$
${\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}dx}$	${\displaystyle ={\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="mwRA">$${\displaystyle ={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}{\frac {b^{2}}{2(ax+b)^{2}}}\right)}$
${\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}dx}$	${\displaystyle ={\frac {1}{a^{3}}}\left(\frac {(ax+b)^{3-n}}{(n-3)}}+{\frac {2b(a+b)^{2-n}}{(n-2)}{\frac {b^{2}(ax+b)^{1-n}}{(n-1)}}\right)\qquad {\mbox{(za }}n\not \in \{1,2,3\}{\mbox{)}}}$

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

${\displaystyle \int {\frac {1}{ax^{2}+bx+c}}dx=}$	${\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(za }}4ac-b^{2}>0{\mbox{)}}}$
	${\displaystyle \frac {2}{\sqrt {b^{24ac}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{24ac}}}={\frac {1}{\sqrt {b^{24ac}}}\ln \left|{\frac {2ax+b\sqrt {b^{24ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{(za }}4ac-b^{2}<0{\mbox{)}}}$
	${\displaystyle -\\frac{b}{2a}\\int\\frac{dx}{ax^2+bx+c}"}}' id="mwjg">$${\displaystyle ={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|ax^{2}+bx+c\right|ax^2+bx+c\\right|+\\frac{2an-bm}{a\\sqrt{4ac-b^2}}\\arctan\\frac{2ax+b}{\\sqrt{4ac-b^2}} \\qquad\\mbox{(za }4ac-b^2>0\\mbox{)}"}}' id="mwmw">$${\displaystyle {\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{(za }}4ac-b^{2}>0{\mbox{)}}}$
	${\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{24ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{24ac}}}\qquad {\mbox{(za }}4ac-b^{2}<0{\mbox{)}}}$
	${\displaystyle {\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|ax^{2}+bx+c\right|\frac {2an-bm}{a(2ax+b)}}\,\,\,\,\,\,\,\,\,\,\qquad {\mbox{(za }}4ac-b^{2}=0{\mbox{)}}}"/>$

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

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

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

Bilo koja racionalna funkcija se može integrirati rabeći gornje jednadžbe i parcijalne razlomke u integriranju, dekompozicijom racionalne funkcije u zbroj funkcija oblika:

${\displaystyle {\frac {ex+f}{\left(ax^{2}+bx+c\right)^{n}}}}$.