Wikipedia, Entziklopedia askea
Ondorengoa alderantzizko funtzio hiperbolikoen integralen zerrenda bat da (jatorrizkoak edo antideribatuak). Integralen zerrenda osatuago nahi baduzu, ikusi integralen zerrenda.
K erabiltzen da integrazio-konstante gisa. Konstante hori zehaztu daiteke soilik integralaren balioa ezaguna baldin bada puntu batean. Horrela, funtzio bakoitzak jatorrizkoen kopuru infinitua dauka.
![{\displaystyle \int \operatorname {arsinh} (a\,x)\,dx=x\,\operatorname {arsinh} (a\,x)-{\frac {\sqrt {a^{2}\,x^{2}+1}}{a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/9660a5387c9c1037f930724cf62661fc7c8d8fd5)
![{\displaystyle \int x\,\operatorname {arsinh} (a\,x)dx={\frac {x^{2}\,\operatorname {arsinh} (a\,x)}{2}}+{\frac {\operatorname {arsinh} (a\,x)}{4\,a^{2}}}-{\frac {x{\sqrt {a^{2}\,x^{2}+1}}}{4\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/7449166f31e0bc0f88a5fc4b31cf7b38c5caaa86)
![{\displaystyle \int x^{2}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{3}\,\operatorname {arsinh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}-2\right){\sqrt {a^{2}\,x^{2}+1}}}{9\,a^{3}}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/2705ab466b204e4ebce149b0ef9257bbe6701f25)
![{\displaystyle \int x^{m}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsinh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {a^{2}\,x^{2}+1}}}\,dx\quad (m\neq -1)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ea3bd92d23f286dcf429192c7b29bd73f6b505c2)
![{\displaystyle \int \operatorname {arsinh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arsinh} (a\,x)^{2}-{\frac {2\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)}{a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/599742d1bac90a967be49f8366fdda0d61e73ce7)
![{\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=x\,\operatorname {arsinh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arsinh} (a\,x)^{n-2}\,dx}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/11c85f05a4f942d35a38a2d54900beced3269ae5)
![{\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arsinh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n+1}}{a(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arsinh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/a42f277f9042e48f58353d76141a3e179ca183bb)
![{\displaystyle \int \operatorname {arcosh} (a\,x)\,dx=x\,\operatorname {arcosh} (a\,x)-{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1f2b056570b80cda2a9fb9bbab5e8a7e69821539)
![{\displaystyle \int x\,\operatorname {arcosh} (a\,x)dx={\frac {x^{2}\,\operatorname {arcosh} (a\,x)}{2}}-{\frac {\operatorname {arcosh} (a\,x)}{4\,a^{2}}}-{\frac {x\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{4\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/8bc6c9cd571db5a0363fae4bf6da360aba966c3e)
![{\displaystyle \int x^{2}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{3}\,\operatorname {arcosh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{9\,a^{3}}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/133791c8a706528b748a1a3b0d960e8820a4fc3f)
![{\displaystyle \int x^{m}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arcosh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}}\,dx\quad (m\neq -1)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4820e27ce27b6cd07b007cb51d02bbbbfaa88f03)
![{\displaystyle \int \operatorname {arcosh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arcosh} (a\,x)^{2}-{\frac {2\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)}{a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/fdc5775a124252f40555b72732f427e3f5c7879b)
![{\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=x\,\operatorname {arcosh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arcosh} (a\,x)^{n-2}\,dx}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/0eba7307717ccdbfd43a8030a47599512da88fdc)
![{\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arcosh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n+1}}{a\,(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arcosh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/fecfb8152bc97787aeffc7e9dec671789620df40)
![{\displaystyle \int \operatorname {artanh} (a\,x)\,dx=x\,\operatorname {artanh} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d128da84241216605a31740738534a36a9ddcc5c)
![{\displaystyle \int x\,\operatorname {artanh} (a\,x)dx={\frac {x^{2}\,\operatorname {artanh} (a\,x)}{2}}-{\frac {\operatorname {artanh} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/158413ab1c446293df57663635969ac3adafd5b7)
![{\displaystyle \int x^{2}\,\operatorname {artanh} (a\,x)dx={\frac {x^{3}\,\operatorname {artanh} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/efa6b789df194782ce381062d00def762aee6de2)
![{\displaystyle \int x^{m}\,\operatorname {artanh} (a\,x)dx={\frac {x^{m+1}\operatorname {artanh} (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/52c29d51a929721aac2215d472c8836d2c5dd478)
![{\displaystyle \int \operatorname {arcoth} (a\,x)\,dx=x\,\operatorname {arcoth} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/300c860298d65f825a6e3be00b636e6c643c3bf2)
![{\displaystyle \int x\,\operatorname {arcoth} (a\,x)dx={\frac {x^{2}\,\operatorname {arcoth} (a\,x)}{2}}-{\frac {\operatorname {arcoth} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/3c5eacefe8cf1db9386709521479c1c79555325e)
![{\displaystyle \int x^{2}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{3}\,\operatorname {arcoth} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/692f7342f2f09715b6957fa11484c632cbc375e2)
![{\displaystyle \int x^{m}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{m+1}\operatorname {arcoth} (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/46b3ea59b6984ba0720668ba293667907bb14ac2)
![{\displaystyle \int \operatorname {arsech} (a\,x)\,dx=x\,\operatorname {arsech} (a\,x)-{\frac {2}{a}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ca16f50bee8aa396d0c0d5aaaaed1104bbd03f11)
![{\displaystyle \int x\,\operatorname {arsech} (a\,x)dx={\frac {x^{2}\,\operatorname {arsech} (a\,x)}{2}}-{\frac {(1+a\,x)}{2\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/3427139170fe2414b73d0f1ad67b2fe924314fd2)
![{\displaystyle \int x^{2}\,\operatorname {arsech} (a\,x)dx={\frac {x^{3}\,\operatorname {arsech} (a\,x)}{3}}\,-\,{\frac {1}{3\,a^{3}}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}\,-\,{\frac {x(1+a\,x)}{6\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}\,+\,K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/86b78c72b66caa49a51b779522e4b4cb0d971f93)
![{\displaystyle \int x^{m}\,\operatorname {arsech} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsech} (a\,x)}{m+1}}\,+\,{\frac {1}{m+1}}\int {\frac {x^{m}}{(1+a\,x){\sqrt {\frac {1-a\,x}{1+a\,x}}}}}\,dx\quad (m\neq -1)}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/51b9d5134d53de17890b5d94f9b80c9a8c4fe3c9)
![{\displaystyle \int \operatorname {arcsch} (a\,x)\,dx=x\,\operatorname {arcsch} (a\,x)+{\frac {1}{a}}\,\operatorname {artanh} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/f59d6be8e1a5eab296dc95a97e05290e3814796e)
![{\displaystyle \int x\,\operatorname {arcsch} (a\,x)dx={\frac {x^{2}\,\operatorname {arcsch} (a\,x)}{2}}+{\frac {x}{2\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/f0697bb3d1b709939666c994262fcc53b2cef7f1)
![{\displaystyle \int x^{2}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{3}\,\operatorname {arcsch} (a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {artanh} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,K}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/21213c1d59a5efffdc8b957d5c06d89ff79935a0)
![{\displaystyle \int x^{m}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{m+1}\operatorname {arcsch} (a\,x)}{m+1}}\,+\,{\frac {1}{a(m+1)}}\int {\frac {x^{m-1}}{\sqrt {{\frac {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/f0b91d7084049349535acf727c9a6e334d0eba26)