من ويكيبيديا، الموسوعة الحرة
هذه قائمة تكاملات الدوال المثلثية العكسية.
الدوال التي تحتوي معكوس الجيب[عدل]
أخذاً بالعلم أن معكوس الجيب (جا−1) =
![{\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddfc7f7ceca71af5b218fa2e6e20ae52c521950)
![{\displaystyle \int \arcsin {\frac {x}{a}}\ dx=x\arcsin {\frac {x}{a}}+{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4df579ac62bc40fcd8580a4ed9d21f7089cb7e56)
![{\displaystyle \int x\arcsin {\frac {x}{a}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}}{4}}\right)\arcsin {\frac {x}{a}}+{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac48380f10104ab20a7407853ea532acff88723d)
![{\displaystyle \int x^{2}\arcsin {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{a}}+{\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b30f9fca41dd2ee9344fb7f24b0dcfe3af6f7c)
![{\displaystyle \int x^{n}\arcsin x\ dx={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n-1}}+n\int x^{n-2}\arcsin x\ dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da23343805dfc989be3082e69a9a68e91e3aaf9b)
![{\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/82baea15e9d96d076cd92339d0f8a08255ac6751)
الدوال التي تحتوي معكوس جيب التمام[عدل]
أخذاً بالعلم أن معكوس جيب التمام (جتا−1) =
![{\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8f0fa234f076adac9b9aef92b59b4b4f3da43d)
![{\displaystyle \int \arccos {\frac {x}{a}}\ dx=x\arccos {\frac {x}{a}}-{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76cf49c41a266aaf43f6e0b2ca093d55c60d5d86)
![{\displaystyle \int x\arccos {\frac {x}{a}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}}{4}}\right)\arccos {\frac {x}{a}}-{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6a5a82133287e6860c58af182dd7a3b1067ef7)
![{\displaystyle \int x^{2}\arccos {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\arccos {\frac {x}{a}}-{\frac {x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00b7b531ebe7529c18497245aee11e4e38ec4cdf)
الدوال التي تحتوي معكوس الظل[عدل]
أخذاً بالعلم أن معكوس الظل (ظا−1) =
![{\displaystyle \int \arctan x\,dx=x\arctan x-{\frac {1}{2}}\ln(1+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63eb0c10e94df17cfac0672b74a0365574f90935)
![{\displaystyle \int \arctan {\big (}{\frac {x}{a}}{\big )}dx=x\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {a}{2}}\ln(1+{\frac {x^{2}}{a^{2}}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75946054034d19e3177c407e313373b8aef9ce94)
![{\displaystyle \int x\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {(a^{2}+x^{2})\arctan {\big (}{\frac {x}{a}}{\big )}-ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e888d77458f27e83bbeaef79b079c53bcfcf208)
![{\displaystyle \int x^{2}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{3}}{3}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({a^{2}+x^{2}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d75614d8781b5dce6c09bfee6de89b8bcc087549)
![{\displaystyle \int x^{n}\arctan {\big (}{\frac {x}{a}}{\big )}dx={\frac {x^{n+1}}{n+1}}\arctan {\big (}{\frac {x}{a}}{\big )}-{\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/5209f4c529d5fc0cfc58f1895efac6211ee79fe3)
الدوال التي تحتوي معكوس ظل التمام[عدل]
أخذاً بالعلم أن معكوس ظل التمام (ظتا−1) =
![{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+{\frac {1}{2}}\ln(1+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffa69f72d107992e6b7c7ba2c2456e3c0e9ca7e)
![{\displaystyle \int \operatorname {arccot} {\frac {x}{a}}\ dx=x\operatorname {arccot} {\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a3bc0613b4ca86964686b5d5673624053056a6)
![{\displaystyle \int x\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {a^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{a}}+{\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a81c46c3bf9236378bd246d70e68c205394021cc)
![{\displaystyle \int x^{2}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\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/7c9bbf3198cccd8c9110a47eb5e38ea4de829e40)
![{\displaystyle \int x^{n}\operatorname {arccot} {\frac {x}{a}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\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/e048d6d8d201060ee510c9a08af2bd9dc7d3ff8d)
الدوال التي تحتوي معكوس القاطع[عدل]
أخذاً بالعلم أن معكوس القاطع (قا−1) =
![{\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c08821ce259a53d03566b4c0d54b3556616cb9e3)
![{\displaystyle \int \operatorname {arcsec} {\frac {x}{a}}\ dx=x\operatorname {arcsec} {\frac {x}{a}}+{\frac {x}{a|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46491d15cd5ec8bcb7f2dddfcf5eaef4309e2487)
![{\displaystyle \int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b36ac9151db90f64be275f4bad3bd133fb1c43)
![{\displaystyle \int x^{n}\operatorname {arcsec} x\ dx={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+[1-n]\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\ dx\right)\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ddbb1b68bbd8854785c4f9425e08803dc092d4)
الدوال التي تحتوي معكوس قاطع التمام[عدل]
أخذاً بالعلم أن معكوس قاطع التمام (قتا−1) =
![{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2}}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea3cea0509e928e590ea967d9edf319ae09ec0f)
![{\displaystyle \int \operatorname {arccsc} {\frac {x}{a}}\ dx=x\operatorname {arccsc} {\frac {x}{a}}+{a}\ln {({\frac {x}{a}}({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1))}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbbd59470c36ef88e3bdb8c2b1949789b5eb2ba)
![{\displaystyle \int x\operatorname {arccsc} {\frac {x}{a}}\ dx={\frac {x^{2}}{2}}\operatorname {arccsc} {\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e957e11113291b7e0f15cfd77c37f456210638)