Цей список інтегралів (первісних функцій) раціональних функцій. Для повнішого списку інтегралів дивись Таблиця інтегралів.
![{\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4cff38f038483530f9f415d8bde05c0a7905e8)
![{\displaystyle \int {\frac {c}{ax+b}}dx={\frac {c}{a}}\ln \left|ax+b\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8bbd89f4b814d85946a365fab61e118148c90e0)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0714a63bf5e4774dc195a7a546845d0b94b3de)
![{\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{(for }}n\not \in \{1,2\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa80e7490d0852cfbd37849838030e8537a8af1)
![{\displaystyle \int {\frac {x^{2}}{ax+b}}dx={\frac {b^{2}\ln(\left|ax+b\right|)}{a^{3}}}+{\frac {ax^{2}-2bx}{2a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7de1f00ca79edbbb44525171413b97efe3fb7ab0)
![{\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}dx={\frac {1}{a^{3}}}\left(ax-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbeb5baad921afae7e0391648a694117826c4cc9)
![{\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 {(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{(for }}n\not \in \{1,2,3\}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0809f9e45e03cd118ad607ba0d30d7c092371a)
![{\displaystyle \int {\frac {1}{x(ax+b)}}dx=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1eab76fea010b410763a2216215d7179e689faf)
![{\displaystyle \int {\frac {1}{x^{2}(ax+b)}}dx=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66e24f2d9d99e478489e520e668e1b73781044f)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37fd81d5e796572f1058cda7aaf9da862a16973a)
![{\displaystyle \int {\frac {1}{x^{2}+a^{2}}}dx={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5604ddf6918bd40bfee55ae49ba24613f036e346)
![{\displaystyle \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}}&{\mbox{(for }}|x|<|a|{\mbox{)}}\\-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}&{\mbox{(for }}|x|>|a|{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1906ef35cba8c97fcbac15a62067bc5544cb3c5)
for
![{\displaystyle \int {\frac {1}{ax^{2}+bx+c}}dx={\begin{cases}{\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}&{\mbox{(for }}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|&{\mbox{(for }}4ac-b^{2}<0{\mbox{)}}\\-{\frac {2}{2ax+b}}&{\mbox{(for }}4ac-b^{2}=0{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fda7ab39edff53724da0f8c50b6ad211c8966a)
||![{\displaystyle ={\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/0e3d08af150772bb43bc9438195724b556eef705)
![{\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\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}}}}&{\mbox{(for }}4ac-b^{2}>0{\mbox{)}}\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}&{\mbox{(for }}4ac-b^{2}<0{\mbox{)}}\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}&{\mbox{(for }}4ac-b^{2}=0{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dcd06dcb1cebb3f90e75b39d8db46f3594fa3a4)
![{\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\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a2659d0200da3cbebeff9bd68045443500142b)
![{\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\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8385b1a9fb6eba6a84fb808c1dd94b8bce6d4d07)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505fd268d136f336322d8635315c3983c3c8aba6)
![{\displaystyle \int {\frac {dx}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\left[\sin({\frac {(2k-1)\pi }{2^{n}}})\arctan[\left(x-\cos({\frac {(2k-1)\pi }{2^{n}}})\right)\csc({\frac {(2k-1)\pi }{2^{n}}})]\right]-{\frac {1}{2^{n}}}\left[\cos({\frac {(2k-1)\pi }{2^{n}}})\ln \left|x^{2}-2x\cos({\frac {(2k-1)\pi }{2^{n}}})+1\right|\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d961dbbe0264f48b1ec1e3ddeb69e9e8d0d83097)
Будь-яка раціональна функція може бути проінтегрована з використанням вищенаведених рівнянь і методу розкладу на прості дроби, тобто декомпозицією раціональної функції в суму функцій вигляду:
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- Двайт Г. Б. Рациональные алгебраические функции — интегралы // Таблицы интегралов и другие математические формулы / пер. с англ. Н. В. Леви ; под ред. К. А. Семендяева. — М. : Наука, 1978. — С. 22-40. (рос.)