Popis integrala racionalnih funkcija

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

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

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

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

$\int {\frac {1}{x(ax+b)}}dx$	$=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|+C$
$\int {\frac {1}{x^{2}(ax+b)}}dx$	$=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|+C$
$\int {\frac {1}{x^{2}(ax+b)^{2}}}dx$	$=-a\left({\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right)+C$
$\int {\frac {1}{x^{2}+a^{2}}}dx$	$={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!+C$
$\int {\frac {1}{x^{2}-a^{2}}}dx=$	${\begin{cases}-{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}+C\qquad {\mbox{(za }}|x|<|a|{\mbox{)}}\,\!\\-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}+C\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}\,\!\end{cases}}$

$\int {\frac {1}{ax^{2}+bx+c}}dx=$

${\begin{cases}{\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C\qquad {\mbox{(za }}4ac-b^{2}>0{\mbox{)}}\\-{\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C\qquad {\mbox{(za }}4ac-b^{2}<0{\mbox{)}}\\-{\frac {2}{2ax+b}}+C\qquad {\mbox{(za }}4ac-b^{2}=0{\mbox{)}}\end{cases}}$

$\int {\frac {x}{ax^{2}+bx+c}}dx$

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{2a}\ln\left|ax^2+bx+c\right|-\frac{b}{2a}\int\frac{dx}{ax^2+bx+c} + C}

$\int {\frac {mx+n}{ax^{2}+bx+c}}dx=$

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \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}} + C \qquad\mbox{(za }4ac-b^2>0\mbox{)} \\ \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}} + C \qquad\mbox{(za }4ac-b^2<0\mbox{)} \\ \frac{m}{2a}\ln\left|ax^2+bx+c\right|-\frac{2an-bm}{a(2ax+b)} + C \,\,\,\,\,\,\,\,\,\, \qquad\mbox{(za }4ac-b^2=0\mbox{)} \end{cases}}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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\,\! + C}

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 + C}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ex + f}{\left(ax^2+bx+c\right)^n}} .