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Følgende er en liste over ubestemte integraler (antideriverte) til uttrykk som involverer de inverse trigonometriske funksjonene. For en liste over integralformler, se lister over integraler.
- De inverse trigonometriske funksjonene er også kjent som de syklometriske funksjonene.
- C brukes for den vilkårlige integrasjonskonstanten som bare kan bestemmes hvis noe om verdien av integralet på noe punkt, er kjent. Derfor har hver funksjon et uendelig antall antideriverte.
- Det er tre vanlige notasjonsmåter for inverse trigonometriske funksjoner. Funksjonen arcsinus, for eksempel, kan skrives som sin−1, asin eller, som brukt på denne siden, arcsin.
- For hver integrasjonsformel for inverse trigonometriske funksjoner nedenfor er det en korresponderende formel i listen over integraler av inverse hyperbolske funksjoner.
Integrasjonsformler for arcsinusfunksjonen[rediger | rediger kilde]
![{\displaystyle \int \arcsin(a\,x)\,dx=x\arcsin(a\,x)+{\frac {\sqrt {1-a^{2}\,x^{2}}}{a}}+C}](https://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/550b06ea0658a8c75e7a817088408692871fee31)
Integrasjonsformler for arccosinusfunksjonen[rediger | rediger kilde]
![{\displaystyle \int \arccos(a\,x)\,dx=x\arccos(a\,x)-{\frac {\sqrt {1-a^{2}\,x^{2}}}{a}}+C}](https://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/46c69f29ac5b26d533e04184f6a964784a89e31c)
Integrasjonsformler for arctangensfunksjonen[rediger | rediger kilde]
![{\displaystyle \int \arctan(a\,x)\,dx=x\arctan(a\,x)-{\frac {\ln \left(a^{2}\,x^{2}+1\right)}{2\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/68cb125ed3fe6492eb89d6d3c390b4f984510988)
![{\displaystyle \int x\arctan(a\,x)\,dx={\frac {x^{2}\arctan(a\,x)}{2}}+{\frac {\arctan(a\,x)}{2\,a^{2}}}-{\frac {x}{2\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/0548577134558e075e17d226af671b1b7da20902)
![{\displaystyle \int x^{2}\arctan(a\,x)\,dx={\frac {x^{3}\arctan(a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}+1\right)}{6\,a^{3}}}-{\frac {x^{2}}{6\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/85c45ce65af282e243e93b1c4ebd7b698999e237)
![{\displaystyle \int x^{m}\arctan(a\,x)\,dx={\frac {x^{m+1}\arctan(a\,x)}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}+1}}\,dx\quad (m\neq -1)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/eb29193d79bf1881799bbe841d537a4c3a75e599)
Integrasjonsformler for arccotangensfunksjonen[rediger | rediger kilde]
![{\displaystyle \int \operatorname {arccot}(a\,x)\,dx=x\operatorname {arccot}(a\,x)+{\frac {\ln \left(a^{2}\,x^{2}+1\right)}{2\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/6eea749c07f4c29c5b89fb09ccb4c6ab218e8045)
![{\displaystyle \int x\operatorname {arccot}(a\,x)\,dx={\frac {x^{2}\operatorname {arccot}(a\,x)}{2}}+{\frac {\operatorname {arccot}(a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/cfa453a7fbe0023e4e13e799db4d242471428bac)
![{\displaystyle \int x^{2}\operatorname {arccot}(a\,x)\,dx={\frac {x^{3}\operatorname {arccot}(a\,x)}{3}}-{\frac {\ln \left(a^{2}\,x^{2}+1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/0fa82bda86eef71253d4951235f025c576ea80c3)
![{\displaystyle \int x^{m}\operatorname {arccot}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arccot}(a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}+1}}\,dx\quad (m\neq -1)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/56f1e9322cc1307bc8a256ce5a0c28c2b93f9dbd)
Integrasjonsformler for arcsecansfunksjonen[rediger | rediger kilde]
![{\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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/eadb5e445fb2a3d92c308f1559678f8b07a1f446)
Integrasjonsformler for arccosecansfunksjonen[rediger | rediger kilde]
![{\displaystyle \int \operatorname {arccsc}(a\,x)\,dx=x\operatorname {arccsc}(a\,x)+{\frac {1}{a}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/c95ca55f64c8a1a28ae3eb45cd14080a82895f77)
![{\displaystyle \int x\operatorname {arccsc}(a\,x)\,dx={\frac {x^{2}\operatorname {arccsc}(a\,x)}{2}}+{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}+C}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d51804fa5bf7c38cd8c384d6913317dccba7a795)
![{\displaystyle \int x^{2}\operatorname {arccsc}(a\,x)\,dx={\frac {x^{3}\operatorname {arccsc}(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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ea11bed035cb26d9a7a742e7f86a748295dfb340)
![{\displaystyle \int x^{m}\operatorname {arccsc}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arccsc}(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://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/324bcc7000abcf407383b0ecde76b7d6076c3019)
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