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The issue of List of integrals of inverse trigonometric functions is of utmost importance today. With the advancement of technology and changes in society, it is crucial to understand the implications and opportunities this topic presents. In this article, we will explore different aspects related to List of integrals of inverse trigonometric functions , from its impact on daily life to its influence on the economy and politics. In addition, we will analyze different perspectives and opinions on the matter, in order to provide a comprehensive and objective vision about List of integrals of inverse trigonometric functions . Read on to discover everything you need to know about this fascinating topic!
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 .
∫
arcsin
(
x
)
d
x
=
x
arcsin
(
x
)
+
1
−
x
2
+
C
{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}
∫
arcsin
(
a
x
)
d
x
=
x
arcsin
(
a
x
)
+
1
−
a
2
x
2
a
+
C
{\displaystyle \int \arcsin(ax)\,dx=x\arcsin(ax)+{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}
∫
x
arcsin
(
a
x
)
d
x
=
x
2
arcsin
(
a
x
)
2
−
arcsin
(
a
x
)
4
a
2
+
x
1
−
a
2
x
2
4
a
+
C
{\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}
∫
x
2
arcsin
(
a
x
)
d
x
=
x
3
arcsin
(
a
x
)
3
+
(
a
2
x
2
+
2
)
1
−
a
2
x
2
9
a
3
+
C
{\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}
∫
x
m
arcsin
(
a
x
)
d
x
=
x
m
+
1
arcsin
(
a
x
)
m
+
1
−
a
m
+
1
∫
x
m
+
1
1
−
a
2
x
2
d
x
,
(
m
≠
−
1
)
{\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)}
∫
arcsin
(
a
x
)
2
d
x
=
−
2
x
+
x
arcsin
(
a
x
)
2
+
2
1
−
a
2
x
2
arcsin
(
a
x
)
a
+
C
{\displaystyle \int \arcsin(ax)^{2}\,dx=-2x+x\arcsin(ax)^{2}+{\frac {2{\sqrt {1-a^{2}x^{2}}}\arcsin(ax)}{a}}+C}
∫
arcsin
(
a
x
)
n
d
x
=
x
arcsin
(
a
x
)
n
+
n
1
−
a
2
x
2
arcsin
(
a
x
)
n
−
1
a
−
n
(
n
−
1
)
∫
arcsin
(
a
x
)
n
−
2
d
x
{\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}
∫
arcsin
(
a
x
)
n
d
x
=
x
arcsin
(
a
x
)
n
+
2
(
n
+
1
)
(
n
+
2
)
+
1
−
a
2
x
2
arcsin
(
a
x
)
n
+
1
a
(
n
+
1
)
−
1
(
n
+
1
)
(
n
+
2
)
∫
arcsin
(
a
x
)
n
+
2
d
x
,
(
n
≠
−
1
,
−
2
)
{\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)}
∫
arccos
(
x
)
d
x
=
x
arccos
(
x
)
−
1
−
x
2
+
C
{\displaystyle \int \arccos(x)\,dx=x\arccos(x)-{\sqrt {1-x^{2}}}+C}
∫
arccos
(
a
x
)
d
x
=
x
arccos
(
a
x
)
−
1
−
a
2
x
2
a
+
C
{\displaystyle \int \arccos(ax)\,dx=x\arccos(ax)-{\frac {\sqrt {1-a^{2}x^{2}}}{a}}+C}
∫
x
arccos
(
a
x
)
d
x
=
x
2
arccos
(
a
x
)
2
−
arccos
(
a
x
)
4
a
2
−
x
1
−
a
2
x
2
4
a
+
C
{\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}
∫
x
2
arccos
(
a
x
)
d
x
=
x
3
arccos
(
a
x
)
3
−
(
a
2
x
2
+
2
)
1
−
a
2
x
2
9
a
3
+
C
{\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}
∫
x
m
arccos
(
a
x
)
d
x
=
x
m
+
1
arccos
(
a
x
)
m
+
1
+
a
m
+
1
∫
x
m
+
1
1
−
a
2
x
2
d
x
,
(
m
≠
−
1
)
{\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)}
∫
arccos
(
a
x
)
2
d
x
=
−
2
x
+
x
arccos
(
a
x
)
2
−
2
1
−
a
2
x
2
arccos
(
a
x
)
a
+
C
{\displaystyle \int \arccos(ax)^{2}\,dx=-2x+x\arccos(ax)^{2}-{\frac {2{\sqrt {1-a^{2}x^{2}}}\arccos(ax)}{a}}+C}
∫
arccos
(
a
x
)
n
d
x
=
x
arccos
(
a
x
)
n
−
n
1
−
a
2
x
2
arccos
(
a
x
)
n
−
1
a
−
n
(
n
−
1
)
∫
arccos
(
a
x
)
n
−
2
d
x
{\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}
∫
arccos
(
a
x
)
n
d
x
=
x
arccos
(
a
x
)
n
+
2
(
n
+
1
)
(
n
+
2
)
−
1
−
a
2
x
2
arccos
(
a
x
)
n
+
1
a
(
n
+
1
)
−
1
(
n
+
1
)
(
n
+
2
)
∫
arccos
(
a
x
)
n
+
2
d
x
,
(
n
≠
−
1
,
−
2
)
{\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)}
∫
arctan
(
x
)
d
x
=
x
arctan
(
x
)
−
ln
(
x
2
+
1
)
2
+
C
{\displaystyle \int \arctan(x)\,dx=x\arctan(x)-{\frac {\ln \left(x^{2}+1\right)}{2}}+C}
∫
arctan
(
a
x
)
d
x
=
x
arctan
(
a
x
)
−
ln
(
a
2
x
2
+
1
)
2
a
+
C
{\displaystyle \int \arctan(ax)\,dx=x\arctan(ax)-{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C}
∫
x
arctan
(
a
x
)
d
x
=
x
2
arctan
(
a
x
)
2
+
arctan
(
a
x
)
2
a
2
−
x
2
a
+
C
{\displaystyle \int x\arctan(ax)\,dx={\frac {x^{2}\arctan(ax)}{2}}+{\frac {\arctan(ax)}{2\,a^{2}}}-{\frac {x}{2\,a}}+C}
∫
x
2
arctan
(
a
x
)
d
x
=
x
3
arctan
(
a
x
)
3
+
ln
(
a
2
x
2
+
1
)
6
a
3
−
x
2
6
a
+
C
{\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}
∫
x
m
arctan
(
a
x
)
d
x
=
x
m
+
1
arctan
(
a
x
)
m
+
1
−
a
m
+
1
∫
x
m
+
1
a
2
x
2
+
1
d
x
,
(
m
≠
−
1
)
{\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)}
∫
arccot
(
x
)
d
x
=
x
arccot
(
x
)
+
ln
(
x
2
+
1
)
2
+
C
{\displaystyle \int \operatorname {arccot}(x)\,dx=x\operatorname {arccot}(x)+{\frac {\ln \left(x^{2}+1\right)}{2}}+C}
∫
arccot
(
a
x
)
d
x
=
x
arccot
(
a
x
)
+
ln
(
a
2
x
2
+
1
)
2
a
+
C
{\displaystyle \int \operatorname {arccot}(ax)\,dx=x\operatorname {arccot}(ax)+{\frac {\ln \left(a^{2}x^{2}+1\right)}{2\,a}}+C}
∫
x
arccot
(
a
x
)
d
x
=
x
2
arccot
(
a
x
)
2
+
arccot
(
a
x
)
2
a
2
+
x
2
a
+
C
{\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}
∫
x
2
arccot
(
a
x
)
d
x
=
x
3
arccot
(
a
x
)
3
−
ln
(
a
2
x
2
+
1
)
6
a
3
+
x
2
6
a
+
C
{\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}
∫
x
m
arccot
(
a
x
)
d
x
=
x
m
+
1
arccot
(
a
x
)
m
+
1
+
a
m
+
1
∫
x
m
+
1
a
2
x
2
+
1
d
x
,
(
m
≠
−
1
)
{\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)}
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
ln
(
|
x
|
+
x
2
−
1
)
+
C
=
x
arcsec
(
x
)
−
arcosh
|
x
|
+
C
{\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}
∫
arcsec
(
a
x
)
d
x
=
x
arcsec
(
a
x
)
−
1
a
arcosh
|
a
x
|
+
C
{\displaystyle \int \operatorname {arcsec}(ax)\,dx=x\operatorname {arcsec}(ax)-{\frac {1}{a}}\,\operatorname {arcosh} |ax|+C}
∫
x
arcsec
(
a
x
)
d
x
=
x
2
arcsec
(
a
x
)
2
−
x
2
a
1
−
1
a
2
x
2
+
C
{\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}
∫
x
2
arcsec
(
a
x
)
d
x
=
x
3
arcsec
(
a
x
)
3
−
arcosh
|
a
x
|
6
a
3
−
x
2
6
a
1
−
1
a
2
x
2
+
C
{\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}
∫
x
m
arcsec
(
a
x
)
d
x
=
x
m
+
1
arcsec
(
a
x
)
m
+
1
−
1
a
(
m
+
1
)
∫
x
m
−
1
1
−
1
a
2
x
2
d
x
,
(
m
≠
−
1
)
{\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)}
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
ln
(
|
x
|
+
x
2
−
1
)
+
C
=
x
arccsc
(
x
)
+
arcosh
|
x
|
+
C
{\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}
∫
arccsc
(
a
x
)
d
x
=
x
arccsc
(
a
x
)
+
1
a
artanh
1
−
1
a
2
x
2
+
C
{\displaystyle \int \operatorname {arccsc}(ax)\,dx=x\operatorname {arccsc}(ax)+{\frac {1}{a}}\,\operatorname {artanh} \,{\sqrt {1-{\frac {1}{a^{2}x^{2}}}}}+C}
∫
x
arccsc
(
a
x
)
d
x
=
x
2
arccsc
(
a
x
)
2
+
x
2
a
1
−
1
a
2
x
2
+
C
{\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}
∫
x
2
arccsc
(
a
x
)
d
x
=
x
3
arccsc
(
a
x
)
3
+
1
6
a
3
artanh
1
−
1
a
2
x
2
+
x
2
6
a
1
−
1
a
2
x
2
+
C
{\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}
∫
x
m
arccsc
(
a
x
)
d
x
=
x
m
+
1
arccsc
(
a
x
)
m
+
1
+
1
a
(
m
+
1
)
∫
x
m
−
1
1
−
1
a
2
x
2
d
x
,
(
m
≠
−
1
)
{\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)}
See also
References
