Daftar integral (antiderivatif) dari ekspresi yang melibatkan fungsi invers trigonometri. Untuk daftar lengkap rumus integral, lihat tabel integral.
- Fungsi invers (= "fungsi kebalikan") trigonometri juga dikenal sebagai "fungsi arc" ("arc functions").
- C digunakan untuk melambangkan konstanta integrasi arbitrari yang hanya dapat ditentukan jika nilai integral pada satu titik tertentu telah diketahui. Jadi setiap fungsi mempunyai antiderivatif yang tak terhingga banyaknya.
- Ada tiga notasi umum untuk fungsi-fungsi invers trigonometri. Fungsi arcsinus, misalnya, dapat ditulis sebagai sin−1, asin, atau, pada halaman ini, arcsin.
- Untuk setiap rumus integrasi fungsi invers trigonometri di bawah ini ada rumus yang bersangkutan dalam daftar integral dari fungsi invers hiperbolik.
Rumus integrasi fungsi arcsinus
![{\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)
Rumus integrasi fungsi arckosinus
![{\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)
Rumus integrasi fungsi arctangen
![{\displaystyle \int \arctan(x)\,dx=x\arctan(x)-{\frac {\ln \left(x^{2}+1\right)}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb38b42688297d90d71c5ebfd567eede2475a32)
![{\displaystyle \int \arctan(a\,x)\,dx=x\arctan(a\,x)-{\frac {\ln \left(a^{2}\,x^{2}+1\right)}{2\,a}}+C}](https://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/eb29193d79bf1881799bbe841d537a4c3a75e599)
Rumus integrasi fungsi arckotangen
![{\displaystyle \int \operatorname {arccot}(x)\,dx=x\operatorname {arccot}(x)+{\frac {\ln \left(x^{2}+1\right)}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c43817a46bd55984e1fe04075f9369d428f17c)
![{\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://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/56f1e9322cc1307bc8a256ce5a0c28c2b93f9dbd)
Rumus integrasi fungsi arcsekan
![{\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3f7e5082b9b9b61c63b54d9e4e3c1a4db532ec)
![{\displaystyle \int \operatorname {arcsec}(ax)\,dx=x\operatorname {arcsec}(ax)-{\frac {1}{a}}\,\operatorname {arcosh} |ax|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c53fe6246297df0234fc323e3d289bfc1e89c3b)
![{\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)
Rumus integrasi fungsi arckosekan
![{\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left|x+{\sqrt {x^{2}-1}}\right|\,+\,C=x\operatorname {arccsc}(x)\,+\,\operatorname {arccosh} (x)\,+\,C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c386426aad2c7e1f61dc902a0c11fbbde27d23)
![{\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://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://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://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://wikimedia.org/api/rest_v1/media/math/render/svg/324bcc7000abcf407383b0ecde76b7d6076c3019)