List of integrals of inverse trigonometric functions
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
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