«Википеди» ирĕклĕ энциклопединчи материал
Аяларах рационаллă функцисен интегралĕсĕн (функцисен умсăнарĕсен) йышне паратпăр.
![{\displaystyle \int (ax+b)^{n}dx={\begin{cases}{\frac {(ax+b)^{n+1}}{a(n+1)}},&n\neq -1\\{\frac {1}{a}}\ln \left|ax+b\right|,&n=-1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c359fd734840de89b47cacd36c6edea3737756b5)
![{\displaystyle \int x(ax+b)^{n}dx={\begin{cases}{\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1},&n\not \in \{-1,-2\}\\{\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|,&n=-1\\{\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|,&n=-2\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97975a8e7e159d337049f414c46601f9afc19613)
![{\displaystyle \int {\frac {x}{(ax+b)^{n}}}dx={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}},\quad n\not \in \{1,2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/950e7a2cb8448b144303f31ebfce9a8674266c8b)
![{\displaystyle \quad \left.-\,{\frac {1}{2^{n}}}\left[\cos \left({\frac {(2k-1)\pi }{2^{n}}}\right)\ln \left|x^{2}-2x\cos \left({\frac {(2k-1)\pi }{2^{n}}}\right)+1\right|\right]\right\}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5deae9573a4f239b1104c54818eb01d22dc67eed)
![{\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)
для ![{\displaystyle n\not \in \{1,2,3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc7492215605a072e29435cd7be7032c3ea6162)
![{\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}{a^{2}x^{2}+b^{2}}}={\frac {1}{ab}}\operatorname {arctg} {\frac {ax}{b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0b1ad94dfbaa2bcb3eb7b5a77f69bfedabce49)
![{\displaystyle \int {\frac {dx}{(x^{2}+a^{2})^{2}}}={\frac {x}{2a^{2}(x^{2}+a^{2})}}+{\frac {1}{2a^{3}}}\operatorname {arctg} {\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93334e34234972d179f530d599801c3d72944ccf)
![{\displaystyle \int {\frac {dx}{(x^{2}+a^{2})^{3}}}={\frac {x}{4a^{2}(x^{2}+a^{2})^{2}}}+{\frac {3x}{8a^{4}(x^{2}+a^{2})}}+{\frac {3}{8a^{5}}}\operatorname {arctg} {\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5338411607bb26da9c6bbb48a959154c64e2eb5)
для ![{\displaystyle |x|<|a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a1b316f0bb57b1b6c0f49c3de731ab72016740)
для ![{\displaystyle |x|>|a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d94ecca9fcce0df024deb10d8fc925e3533845a)
для ![{\displaystyle 4ac-b^{2}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518c207fb90ddb262637565bdc76184f6dfbfd45)
для ![{\displaystyle 4ac-b^{2}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01a3d0b1553bd138bb90872da9f732e805a4919c)
![{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}\qquad {\mbox{(for }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/726d9bf098c53d25f01e4eea4da42e77c1cbb23c)
![{\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)
для ![{\displaystyle 4ac-b^{2}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518c207fb90ddb262637565bdc76184f6dfbfd45)
для ![{\displaystyle 4ac-b^{2}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01a3d0b1553bd138bb90872da9f732e805a4919c)
для ![{\displaystyle 4ac-b^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e535f9ecc58dcff0dbb6650fe239fc2d8ffab9d2)
![{\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/31e156eb97b60497f040735b6dc22177b8b961bd)
![{\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/b9782f61ede00956cdbd28005175e387e3ec6616)
![{\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)
- Кĕнекесем
- Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
- Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
- D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4
- Интегралсен таблицисем
- Интегралсене шутлани
Шаблон:Интегралсен йышĕсен библиографийĕ
Шаблон:Интегралсен йышĕсем