Jen estas listo de nedifinitaj integraloj (malderivaĵoj) de racionalaj funkcioj.
![{\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{(por }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fea92b471ed0673a8c93f9987cc4bc76636b3d69)
![{\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e05016a6b4ac0605b2448dccee82cf7bdeba8e)
![{\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{(por }}n\not \in \{1,2\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f615c82d152660fc8fb02d541aced8950801d47)
![{\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d50d6a0cd0856b7a0fcce5e3de3bf4a8538f5f3d)
![{\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2cfab1a428a74873f579bf819a23bc0b65db8f0)
![{\displaystyle \int {\frac {x}{(ax+b)^{n}}}dx={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{(por }}n\not \in \{1,2\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b99ed0e7f9795dc37c3b24c9c27e1c2803b79665)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24dc58d1d088115d91ef731f8f9b39230c8ccabb)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbe66e3be09ccdb6ee4d48a904df395bec5bdd0)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f6ffff73c920a3f645476ced050b1a37f550f9)
![{\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}dx={\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{(por }}n\not \in \{1,2,3\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ca8624745b2d0674b7badf2b18347ae0b249c2)
![{\displaystyle \int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/912bdc0ba5ffa96e4bc04612d20c000c85c25bb1)
![{\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3411218647b4eb104c943ac340dd39fbb856adfe)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b02fe548ffe70b93e222de3436a2863ac6487da4)
![{\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6462ede38d35cbf1ca1f8c8fae0ef5d2ef2e9e)
![{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\mbox{(por }}|x|<|a|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc423c63d90caceaa018e2b8d9ad0f9f91b8b10)
![{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\mbox{(por }}|x|>|a|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/807b094297015e5427bce3ed6f01ec842fefeb7e)
![{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(por }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/567cac11ef6b8365e16026450ff9cde40a50196a)
![{\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^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{(por }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7341d922f39b6885145c6f49a6f03f950a0de1e6)
![{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}\qquad {\mbox{(por }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/912425382732a86eea032d7575608004388cde2b)
![{\displaystyle \int {\frac {x}{ax^{2}+bx+c}}dx={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742ca6fd0f058dcd9e83be78b566a387df2ad396)
![{\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{(por }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7f7a2cbe6c3c37de3616cb34f0c5b5945e38ee)
![{\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{(por }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/934116bc3e0ac10eebb1ef561a577742ebd66740)
![{\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}\qquad {\mbox{(por }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8d4018da43dafc697b7c2145cc11c408bc51ed)
![{\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}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c079949856bc293bcbdbe3de566cf9359a42a3)
![{\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 {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f63da0c0a982b71c1ba5a17ad1a267c1c60a781)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c358c1f184af0f3b79eb1e30dd5bb94ca47304)
Ĉiu racionala funkcio povas esti integralita uzante pli suprajn egalaĵojn kaj partajn frakciojn en integralado, per malkomponado la funkcio en sumon de funkcioj de tipo
.