本項は、無理関数の原始関数の一覧である。さらに完全な原始関数の一覧は、原始関数の一覧を参照のこと。本項で、積分定数は簡便のために省略している。
を含む無理関数
![{\displaystyle \int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/057de84eaf02efa9ffba665a6094ba3439632d8a)
![{\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {3}{8}}a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/459e66df823c1883944d6c47d2b24a307d1726fd)
![{\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69530dada3fe7e20bfdd87fba4de51783fc31183)
![{\displaystyle \int xr\;dx={\frac {r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3d78f3e68e641a9b0aab864274113ec4ec57eb)
![{\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d723e526aaa761351215a402932ce3cf3dd6b73d)
![{\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6583ead420aa74655131aaf0ed82cbb82e36157)
![{\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697087c789855ff9e55918ad41818251d5b940a7)
![{\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d0c710fe67b89173e00e0f8f877b089029473d)
![{\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c286c93cf9b39345e6c65e118526cf6b0ebba341)
![{\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca1e44ac2d54d893799548feac05e8684a650d4)
![{\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48c6fc659e5a2eb4a86664f238b542283bc1ee2)
![{\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/235dc49fb2ec51fc06ad3c0ed644b5032eed78e5)
![{\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61b0bbd8eb25830f6ff87d1818574f783d0ce222)
![{\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd4bfbe0c7d57a09ac1901c6522cb7e8327d48)
![{\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982e3258c17a21f19d9bdfa66ccbb2e675a32564)
![{\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37516c175afbb2ba492a368426ca0a68db63935)
![{\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b71bf68287d1ad850f92cd2263ed8db19dac6cb)
![{\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a99fe55679d157911cd614e9ca510e54d729d42)
![{\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d35f2538f5e4bbde4837e0642fdceb6e151c5062)
![{\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0505fdb3e13b7e0704f61b5285649e041ec8d66)
![{\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a78aad92f1fd09df4f4c33d5b28081aec09c45)
![{\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e903aacbfd829184b3eeb9233ce14ac236d1e6b)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3156a33834a19e192af67f3a54da6ddbe4ce1)
![{\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efc8a15f3072555233b2781dc4c398780f904e6)
![{\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsinh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4c3c8dc6bbb70a6c76f4d2add4aa72c7f02c86)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7217f767426c8ffb2d25674cf08d8d9c1a30035e)
を含む無理関数
を前提とする。
の場合は次節:
![{\displaystyle \int s\;dx={\frac {1}{2}}\left(xs-a^{2}\ln(x+s)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38bac083c48d1b0bcf255ad161c8de83455912a3)
![{\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ee984a361d44c45466b1c0d5bdddc61425ca6c)
![{\displaystyle \int {\frac {s\;dx}{x}}=s-a\arccos \left|{\frac {a}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1173c728eef3604bfec78fa8b1dc800905f0cb1)
![{\displaystyle \int {\frac {dx}{s}}=\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fed05f1468654080f2fffafe9dea36c935ec339)
ここで、
,
の正の値を採用する。
![{\displaystyle \int {\frac {x\;dx}{s}}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17011581cfa3d9e53920f319ac77a75013dec77)
![{\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f2035386db8b8331fafbbb54ba1c1cf994fb5d)
![{\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcfb12b7156194900353c9def5d1e095149ce99)
![{\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db234e0dc43716a52f17f53c9b79e30303aaf18)
![{\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/808ffdc762b49c281071fd22b77938afa3e0cd5a)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819c367c3f01849bfcfa4f310a2e2fc8d4630d2e)
![{\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25191a1c6fa21c48b6b89f476b96e8c7fa54a541)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755124bfe54faeb5f7029ee29993b42e0335da23)
![{\displaystyle \int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95143cd1e023e990284ba780dcf4d8d6089ac254)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ba248b7ebdb39ad25a8b8e98996b7b7994162e)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d29db4a76fe21b51b57e5e47e8136b88280fd5)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56fdbdff0c7f9f2437cd5c03b5772496597a6ab)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ac2308ed60218f4e246884e5783df8e2ebc54)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0c92cdcb44ecfe7179711341a1964ef2a0782f)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea4b7b2973dd3e2affa09931a8bf41316161f1)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/239ff6c41a3342440c712b9f0c4940e8e6a000d2)
を含む無理関数
![{\displaystyle \int u\;dx={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c38e2826766e3ba4feca78db794b4176f5cb96d)
![{\displaystyle \int xu\;dx=-{\frac {1}{3}}u^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eed0f3cd0c0d48a63c480f4a1e2eb4da5951fa43)
![{\displaystyle \int x^{2}u\;dx=-{\frac {x}{4}}u^{3}+{\frac {a^{2}}{8}}(xu+a^{2}\arcsin {\frac {x}{a}})\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53c9fdb39e3b4b1702f78a5250760125a0ba61ef)
![{\displaystyle \int {\frac {u\;dx}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85f145be76d4f31f6e306b127e3e8dd2bf0c30e)
![{\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacefb8fb4df4a4cd90b7e24706b39ce53a66023)
![{\displaystyle \int {\frac {x^{2}\;dx}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d027d6e3073d768e6d434e22e9e55305b7857cb)
![{\displaystyle \int u\;dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0929015a0ba01414ef650e427435f8ad3e7942f9)
![{\displaystyle \int {\frac {x}{u}}\;dx=-u\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820821887dc319768b2e8ba1869e7d8601ed821c)
を含む無理関数
(ax2 + bx + c)は、任意のp, qに対して(px + q)2より小さくなることはないということを前提とする。
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(for }}a>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3add2ea6d0465dc1f8f3de7193104f8bad5b7a4b)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837e1ab91fe899e88b6f3c4b13666e8697eb3013)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556efdcbf8ee92bfb28e48482149c769d9125052)
![{\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left|2ax+b\right|<{\sqrt {b^{2}-4ac}}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4017a60a8edb505fe2149a772d5e231c1f1ed9)
![{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/086934b294e8b53bebe7b53241bad912f4212dee)
![{\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6887eff55e44af7ed031fa1d919d3de3f379a90b)
![{\displaystyle \int {\frac {dx}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {dx}{R^{2n-1}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd19c82abfd6ab01d93cc3f2691059d4b4915c)
![{\displaystyle \int {\frac {x}{R}}\;dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6880a823009f7d8fe7e2579fa228324ef0d15f03)
![{\displaystyle \int {\frac {x}{R^{3}}}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96545813e77cd2cc46ac4034b607a9457df212a0)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6de2ef912d111ee50377af9daaf84b45633498)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4c2981ae219f385ff1e91f03659b9d15b9a836)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9d34b7cd7ee7609559c1c8e42a3c032594fb4)
を含む無理関数
![{\displaystyle \int S{dx}={\frac {2S^{3}}{3a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf2b6eade9d02ac453412bf9b92ae1f34f17166)
![{\displaystyle \int {\frac {dx}{S}}={\frac {2S}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6458c8a42c9321a984f3efc64b9045abd706ca70)
![{\displaystyle \int {\frac {dx}{xS}}={\begin{cases}-{\frac {2}{\sqrt {b}}}\mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\-{\frac {2}{\sqrt {b}}}\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\{\frac {2}{\sqrt {-b}}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be3dfc654adbecce24d2d6ae77ac38ec8ef47458)
![{\displaystyle \int {\frac {S}{x}}\,dx={\begin{cases}2\left(S-{\sqrt {b}}\,\mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\2\left(S-{\sqrt {b}}\,\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7d0719e988fd577207e461d0e4254cdf83542d)
![{\displaystyle \int {\frac {x^{n}}{S}}dx={\frac {2}{a(2n+1)}}\left(x^{n}S-bn\int {\frac {x^{n-1}}{S}}dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09899a28471c83f4cac79cd12aae623267cd1559)
![{\displaystyle \int x^{n}Sdx={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{n-1}Sdx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/360f03a6794a0b9f164c980d67792e76cade5c18)
![{\displaystyle \int {\frac {1}{x^{n}S}}dx=-{\frac {1}{b(n-1)}}\left({\frac {S}{x^{n-1}}}+\left(n-{\frac {3}{2}}\right)a\int {\frac {dx}{x^{n-1}S}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbc75cbf40bd6cccafd547daf97834987a3d9ff)
出典
- S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)