Վիքիպեդիայից՝ ազատ հանրագիտարանից
Հակադարձ եռանկյունաչափական ֆունկցիաների ինտեգրալների ցանկ, ստորև ներկայացված են արկսինուս, արկկոսինուս, արկտանգենս, արկկոտանգենս, արկսեկանս, արկկոսեկանս ֆունկցիաների ինտեգրալների ցանկերը։
![{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aca6ab38a9c44c197ca561b42f8584c1715d70a)
![{\displaystyle \int \arcsin(a\,x)\,dx=x\arcsin(a\,x)+{\frac {\sqrt {1-a^{2}\,x^{2}}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a03845c8d076e52a9ae53660833bd6489716be)
![{\displaystyle \int x\arcsin(a\,x)\,dx={\frac {x^{2}\arcsin(a\,x)}{2}}-{\frac {\arcsin(a\,x)}{4\,a^{2}}}+{\frac {x{\sqrt {1-a^{2}\,x^{2}}}}{4\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4d96e527bab715d1c9f027b7df550880b0677e)
![{\displaystyle \int x^{2}\arcsin(a\,x)\,dx={\frac {x^{3}\arcsin(a\,x)}{3}}+{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {1-a^{2}\,x^{2}}}}{9\,a^{3}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90d3faa3931e2f586a28e2d1cdedaf003cedac42)
![{\displaystyle \int x^{m}\arcsin(a\,x)\,dx={\frac {x^{m+1}\arcsin(a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}\,x^{2}}}}\,dx\quad (m\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba63798b882b4faede4591c217dc075a6039b88a)
![{\displaystyle \int \arcsin(a\,x)^{2}\,dx=-2\,x+x\arcsin(a\,x)^{2}+{\frac {2{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839cb4a59769916952c362da40e5c894b75171cf)
![{\displaystyle \int \arcsin(a\,x)^{n}\,dx=x\arcsin(a\,x)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)^{n-1}}{a}}\,-\,n\,(n-1)\int \arcsin(a\,x)^{n-2}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84e79704429b771b601837fd591fce1484285a18)
![{\displaystyle \int \arcsin(a\,x)^{n}\,dx={\frac {x\arcsin(a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arcsin(a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/550b06ea0658a8c75e7a817088408692871fee31)
![{\displaystyle \int \cos ^{n}x\arcsin x\,dx=\left(x^{n^{2}+1}\arccos x+{\frac {x^{n}{\sqrt {1-x^{4}}}-nx^{n^{2}-1}\arccos x}{n^{2}-1}}+n\int x^{n^{2}-2}\arccos x\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9102f4684a8a0fcb20614b3a47a32e563276069)
![{\displaystyle \int \arccos(x)\,dx=x\arccos(x)-{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f61e0b80d0d4befdf52e3945d34279fdf4751849)
![{\displaystyle \int \arccos(a\,x)\,dx=x\arccos(a\,x)-{\frac {\sqrt {1-a^{2}\,x^{2}}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f444889aedee249b76de7d229b0a3ce1dd4f73da)
![{\displaystyle \int x\arccos(a\,x)\,dx={\frac {x^{2}\arccos(a\,x)}{2}}-{\frac {\arccos(a\,x)}{4\,a^{2}}}-{\frac {x{\sqrt {1-a^{2}\,x^{2}}}}{4\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb69fb1e8a9db63732fff5a62639b1eef5096bf)
![{\displaystyle \int x^{2}\arccos(a\,x)\,dx={\frac {x^{3}\arccos(a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {1-a^{2}\,x^{2}}}}{9\,a^{3}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd41df8260d3c15de3ba8aa27fad72d1909bcd5)
![{\displaystyle \int x^{m}\arccos(a\,x)\,dx={\frac {x^{m+1}\arccos(a\,x)}{m+1}}\,+\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {1-a^{2}\,x^{2}}}}\,dx\quad (m\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ee494e7a856c52fbc51d4c735a524fe528bcdb)
![{\displaystyle \int \arccos(a\,x)^{2}\,dx=-2\,x+x\arccos(a\,x)^{2}-{\frac {2{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec588f3ccc7f75c3b427ed5dfb24c91500555892)
![{\displaystyle \int \arccos(a\,x)^{n}\,dx=x\arccos(a\,x)^{n}\,-\,{\frac {n{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)^{n-1}}{a}}\,-\,n\,(n-1)\int \arccos(a\,x)^{n-2}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8328254b68b25e1aa99dfe77f0b1ef475d6d3c0)
![{\displaystyle \int \arccos(a\,x)^{n}\,dx={\frac {x\arccos(a\,x)^{n+2}}{(n+1)\,(n+2)}}\,-\,{\frac {{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arccos(a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46c69f29ac5b26d533e04184f6a964784a89e31c)
![{\displaystyle \int \operatorname {arctg} \,x\,dx=x\,\operatorname {arctg} \,x-{\frac {1}{2}}\ln(1+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a076e9c5d9ff81f6f854868d893913a6f4004e)
![{\displaystyle \int \operatorname {arctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d06c74e050978f95c89277608339bf0bb3e9c7a7)
![{\displaystyle \int x\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {(a^{2}+x^{2})\,\operatorname {arctg} \,{\frac {x}{a}}-ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb67299aefd127af760fd7fa28bdd91329069b7)
![{\displaystyle \int x^{2}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c56a0658c732ee1b187480839de0b0bb97cd41de)
![{\displaystyle \int x^{n}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1}}\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\,dx,\quad n\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdaf2eece71b4de8057920181619ad3e4d908e8)
![{\displaystyle \int \operatorname {arcctg} \,x\,dx=x\,\operatorname {arcctg} \,x+{\frac {1}{2}}\ln(1+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76d9dab876aa93c6537a9caaa539d6b6385cc140)
![{\displaystyle \int \operatorname {arcctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe1f13825a8e6d608ddee96101d64274229c7ff)
![{\displaystyle \int x\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {a^{2}+x^{2}}{2}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7804082adcec7d027d48cdbfa70529443948b2)
![{\displaystyle \int x^{2}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a^{2}+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d80d5950c5828d99d683c5925d7a03ae9f4bd55)
![{\displaystyle \int x^{n}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1}}\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^{2}+x^{2}}}\,dx,\quad n\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0e724eebe49c8e520c34a9ce0bf9a2b892c748)
![{\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)-\operatorname {arctan} \,{\sqrt {1-{\frac {1}{x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a8d0f7cb24ba16e91f576c9b076728ccfd3663)
![{\displaystyle \int \operatorname {arcsec}(a\,x)\,dx=x\operatorname {arcsec}(a\,x)-{\frac {1}{a}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/136b105d97768413904d863c00d86deaf14932bb)
![{\displaystyle \int x\operatorname {arcsec}(a\,x)\,dx={\frac {x^{2}\operatorname {arcsec}(a\,x)}{2}}-{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c67214eeacec568312d5737961c7bf6b937e8373)
![{\displaystyle \int x^{2}\operatorname {arcsec}(a\,x)\,dx={\frac {x^{3}\operatorname {arcsec}(a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}\,-\,{\frac {x^{2}}{6\,a}}{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}\,+\,C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1eb18ab660b77f3962fb56a8a074cdc574aed4)
![{\displaystyle \int x^{m}\operatorname {arcsec}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arcsec}(a\,x)}{m+1}}\,-\,{\frac {1}{a\,(m+1)}}\int {\frac {x^{m-1}}{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}}\,dx\quad (m\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eadb5e445fb2a3d92c308f1559678f8b07a1f446)
![{\displaystyle \int \operatorname {arccosec} \,x\,dx=x\,\operatorname {arccosec} \,x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52bc5e48ad86187c8c57336b9bce8366285c28e9)
![{\displaystyle \int \,\operatorname {arccosec} \,{\frac {x}{a}}\,dx=x\,\operatorname {arccosec} \,{\frac {x}{a}}+{a}\ln {\left({\frac {x}{a}}\left({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1\right)\right)}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aabc61afd9ec31b9ab05792bf0a685e1d1c258ca)
![{\displaystyle \int x\,\operatorname {arccosec} \,{\frac {x}{a}}\,dx={\frac {x^{2}}{2}}\,\operatorname {arccosec} \,{\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0c643f03c8e129bd30b84cf623d3099fd2ac3f)
- Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений (4-е издание). 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