Trigonometry |
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![](//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/250px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png) |
- Functions (sin, cos, tan, inverse)
- Generalized trigonometry
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Reference |
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Laws and theorems |
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- Sines
- Cosines
- Tangents
- Cotangents
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Calculus |
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Mathematicians |
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The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals.
- The inverse trigonometric functions are also known as the "arc functions".
- C is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.
- There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin−1, asin, or, as is used on this page, arcsin.
- For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions.
Arcsine function integration formulas
![{\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(ax)\,dx=x\arcsin(ax)+{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a785a88585c44dc18b3af46048c27fdd518a86)
![{\displaystyle \int x\arcsin(ax)\,dx={\frac {x^{2}\arcsin(ax)}{2}}-{\frac {\arcsin(ax)}{4\,a^{2}}}+{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfad0632fd301b4e3ac125353ec6fba8f66f693)
![{\displaystyle \int x^{2}\arcsin(ax)\,dx={\frac {x^{3}\arcsin(ax)}{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/7c7d3f6e0213b17a4b8f835829c8e6e4e4a9be70)
![{\displaystyle \int x^{m}\arcsin(ax)\,dx={\frac {x^{m+1}\arcsin(ax)}{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/7712f2b040354232fd0cb6514597bca22168916b)
![{\displaystyle \int \arcsin(ax)^{2}\,dx=-2x+x\arcsin(ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fadc0494d7eb98326f97ec39fcdeb242306f66ec)
![{\displaystyle \int \arcsin(ax)^{n}\,dx=x\arcsin(ax)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)^{n-1}}{a}}\,-\,n\,(n-1)\int \arcsin(ax)^{n-2}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57c24312837fb5a75b7f5d0c7097f617b54cd150)
![{\displaystyle \int \arcsin(ax)^{n}\,dx={\frac {x\arcsin(ax)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arcsin(ax)^{n+2}\,dx\,,\quad (n\neq -1,-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f17649a89baff53e49ffe32308132f590860848)
Arccosine function integration formulas
![{\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(ax)\,dx=x\arccos(ax)-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cede68d5c51283498d175b2ed41afbe549ba8037)
![{\displaystyle \int x\arccos(ax)\,dx={\frac {x^{2}\arccos(ax)}{2}}-{\frac {\arccos(ax)}{4\,a^{2}}}-{\frac {x{\sqrt {1-a^{2}x^{2}}}}{4\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45d6822ad6dbc146b379ca1354db37b50e61f6a0)
![{\displaystyle \int x^{2}\arccos(ax)\,dx={\frac {x^{3}\arccos(ax)}{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/fbde0f474037b2e8fae1fc417f6f18b33eba1885)
![{\displaystyle \int x^{m}\arccos(ax)\,dx={\frac {x^{m+1}\arccos(ax)}{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/7438ad4d7a10792f092e398b2daad2dfc6e1ac8d)
![{\displaystyle \int \arccos(ax)^{2}\,dx=-2x+x\arccos(ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos(ax)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/759ffa89a0eb49f816b976ef8da0d21a631f4206)
![{\displaystyle \int \arccos(ax)^{n}\,dx=x\arccos(ax)^{n}\,-\,{\frac {n{\sqrt {1-a^{2}x^{2}}}\arccos(ax)^{n-1}}{a}}\,-\,n\,(n-1)\int \arccos(ax)^{n-2}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b91210b75bbee0faa6296d814dae0b1eb5471c)
![{\displaystyle \int \arccos(ax)^{n}\,dx={\frac {x\arccos(ax)^{n+2}}{(n+1)\,(n+2)}}\,-\,{\frac {{\sqrt {1-a^{2}x^{2}}}\arccos(ax)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arccos(ax)^{n+2}\,dx\,,\quad (n\neq -1,-2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16589b4f0d52354588391f38d892a14bb8e76258)
Arctangent function integration formulas
![{\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(ax)\,dx=x\arctan(ax)-{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed844c21c2ce8db5e2229ffc691e4b667f7d212)
![{\displaystyle \int x\arctan(ax)\,dx={\frac {x^{2}\arctan(ax)}{2}}+{\frac {\arctan(ax)}{2\,a^{2}}}-{\frac {x}{2\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd5087e52fe34e0b35d183d5bcdac191a43f8a49)
![{\displaystyle \int x^{2}\arctan(ax)\,dx={\frac {x^{3}\arctan(ax)}{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/44698b0de77ff1a21a9f90a351037fd5f9314615)
![{\displaystyle \int x^{m}\arctan(ax)\,dx={\frac {x^{m+1}\arctan(ax)}{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/7e00614e4744e0f71dd6db23fe0216f0b11c3d0c)
Arccotangent function integration formulas
![{\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}(ax)\,dx=x\operatorname {arccot}(ax)+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06be47b54b9747038efd1d690ed9728b85bda923)
![{\displaystyle \int x\operatorname {arccot}(ax)\,dx={\frac {x^{2}\operatorname {arccot}(ax)}{2}}+{\frac {\operatorname {arccot}(ax)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6f29975daafb00d3ae1e5acde1bb73703ea7b3)
![{\displaystyle \int x^{2}\operatorname {arccot}(ax)\,dx={\frac {x^{3}\operatorname {arccot}(ax)}{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/af110a4cafbe45ebf0321633cdbf6031656671ab)
![{\displaystyle \int x^{m}\operatorname {arccot}(ax)\,dx={\frac {x^{m+1}\operatorname {arccot}(ax)}{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/e06cc11ff1df12ab5c0de4366e69189aa5be1b28)
Arcsecant function integration formulas
![{\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}(ax)\,dx={\frac {x^{2}\operatorname {arcsec}(ax)}{2}}-{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7343de7d0b7d04a41ca4d7c424f66f56ba32193f)
![{\displaystyle \int x^{2}\operatorname {arcsec}(ax)\,dx={\frac {x^{3}\operatorname {arcsec}(ax)}{3}}\,-\,{\frac {\operatorname {arcosh} |ax|}{6\,a^{3}}}\,-\,{\frac {x^{2}}{6\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}\,+\,C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/620fdaa97e513a65c6257712eaf1f14615db67ca)
![{\displaystyle \int x^{m}\operatorname {arcsec}(ax)\,dx={\frac {x^{m+1}\operatorname {arcsec}(ax)}{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/50526221bf7f8674ba1ed07eb45f9156d5cbd774)
Arccosecant function integration formulas
![{\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arccsc}(x)\,+\,\operatorname {arcosh} |x|\,+\,C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8945ddd724e2e41dc277c3512d83191d6ddccce)
![{\displaystyle \int \operatorname {arccsc}(ax)\,dx=x\operatorname {arccsc}(ax)+{\frac {1}{a}}\,\operatorname {artanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b842cc0dbaf73eb13ab43c61a06947da37e41e)
![{\displaystyle \int x\operatorname {arccsc}(ax)\,dx={\frac {x^{2}\operatorname {arccsc}(ax)}{2}}+{\frac {x}{2\,a}}{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60383644d9ffb3f0612f43464f1a03157a51fbad)
![{\displaystyle \int x^{2}\operatorname {arccsc}(ax)\,dx={\frac {x^{3}\operatorname {arccsc}(ax)}{3}}\,+\,{\frac {1}{6\,a^{3}}}\,\operatorname {artanh} \,{\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/b7e8fe65dd11fe8525f0f8d95484d71c0e56aa59)
![{\displaystyle \int x^{m}\operatorname {arccsc}(ax)\,dx={\frac {x^{m+1}\operatorname {arccsc}(ax)}{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/725c51d7015822a8bcb2f9b4161e7fd19c60e84a)
See also
References