Daftar integral dari fungsi invers trigonometri

Daftar integral (antiderivatif) dari ekspresi yang melibatkan fungsi invers trigonometri. Untuk daftar lengkap rumus integral, lihat tabel integral.

  • Fungsi invers (= "fungsi kebalikan") trigonometri juga dikenal sebagai "fungsi arc" ("arc functions").
  • C digunakan untuk melambangkan konstanta integrasi arbitrari yang hanya dapat ditentukan jika nilai integral pada satu titik tertentu telah diketahui. Jadi setiap fungsi mempunyai antiderivatif yang tak terhingga banyaknya.
  • Ada tiga notasi umum untuk fungsi-fungsi invers trigonometri. Fungsi arcsinus, misalnya, dapat ditulis sebagai sin−1, asin, atau, pada halaman ini, arcsin.
  • Untuk setiap rumus integrasi fungsi invers trigonometri di bawah ini ada rumus yang bersangkutan dalam daftar integral dari fungsi invers hiperbolik.

Rumus integrasi fungsi arcsinus

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(a\,x)\,dx=x\arcsin(a\,x)+{\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(a\,x)\,dx={\frac {x^{2}\arcsin(a\,x)}{2}}-{\frac {\arcsin(a\,x)}{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(a\,x)\,dx={\frac {x^{3}\arcsin(a\,x)}{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(a\,x)\,dx={\frac {x^{m+1}\arcsin(a\,x)}{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(a\,x)^{2}\,dx=-2\,x+x\arcsin(a\,x)^{2}+{\frac {2{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)}{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(a\,x)^{n}\,dx=x\arcsin(a\,x)^{n}\,+\,{\frac {n{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)^{n-1}}{a}}\,-\,n\,(n-1)\int \arcsin(a\,x)^{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(a\,x)^{n}\,dx={\frac {x\arcsin(a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {1-a^{2}\,x^{2}}}\arcsin(a\,x)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arcsin(a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}

Rumus integrasi fungsi arckosinus

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(a\,x)\,dx=x\arccos(a\,x)-{\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(a\,x)\,dx={\frac {x^{2}\arccos(a\,x)}{2}}-{\frac {\arccos(a\,x)}{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(a\,x)\,dx={\frac {x^{3}\arccos(a\,x)}{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(a\,x)\,dx={\frac {x^{m+1}\arccos(a\,x)}{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(a\,x)^{2}\,dx=-2\,x+x\arccos(a\,x)^{2}-{\frac {2{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)}{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(a\,x)^{n}\,dx=x\arccos(a\,x)^{n}\,-\,{\frac {n{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)^{n-1}}{a}}\,-\,n\,(n-1)\int \arccos(a\,x)^{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(a\,x)^{n}\,dx={\frac {x\arccos(a\,x)^{n+2}}{(n+1)\,(n+2)}}\,-\,{\frac {{\sqrt {1-a^{2}\,x^{2}}}\arccos(a\,x)^{n+1}}{a\,(n+1)}}\,-\,{\frac {1}{(n+1)\,(n+2)}}\int \arccos(a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}

Rumus integrasi fungsi arctangen

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(a\,x)\,dx=x\arctan(a\,x)-{\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(a\,x)\,dx={\frac {x^{2}\arctan(a\,x)}{2}}+{\frac {\arctan(a\,x)}{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(a\,x)\,dx={\frac {x^{3}\arctan(a\,x)}{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(a\,x)\,dx={\frac {x^{m+1}\arctan(a\,x)}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}+1}}\,dx\quad (m\neq -1)}

Rumus integrasi fungsi arckotangen

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}(a\,x)\,dx=x\operatorname {arccot}(a\,x)+{\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}(a\,x)\,dx={\frac {x^{2}\operatorname {arccot}(a\,x)}{2}}+{\frac {\operatorname {arccot}(a\,x)}{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}(a\,x)\,dx={\frac {x^{3}\operatorname {arccot}(a\,x)}{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}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arccot}(a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}+1}}\,dx\quad (m\neq -1)}

Rumus integrasi fungsi arcsekan

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}(a\,x)\,dx={\frac {x^{2}\operatorname {arcsec}(a\,x)}{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 1 6 a 3 arctanh 1 1 a 2 x 2 x 2 6 a 1 1 a 2 x 2 + C {\displaystyle \int x^{2}\operatorname {arcsec}(a\,x)\,dx={\frac {x^{3}\operatorname {arcsec}(a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {arctanh} \,{\sqrt {1-{\frac {1}{a^{2}\,x^{2}}}}}\,-\,{\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}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arcsec}(a\,x)}{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)}

Rumus integrasi fungsi arckosekan

arccsc ( x ) d x = x arccsc ( x ) + ln | x + x 2 1 | + C = x arccsc ( x ) + arccosh ( x ) + C {\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left|x+{\sqrt {x^{2}-1}}\right|\,+\,C=x\operatorname {arccsc}(x)\,+\,\operatorname {arccosh} (x)\,+\,C}
arccsc ( a x ) d x = x arccsc ( a x ) + 1 a arctanh 1 1 a 2 x 2 + C {\displaystyle \int \operatorname {arccsc}(a\,x)\,dx=x\operatorname {arccsc}(a\,x)+{\frac {1}{a}}\,\operatorname {arctanh} \,{\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}(a\,x)\,dx={\frac {x^{2}\operatorname {arccsc}(a\,x)}{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 arctanh 1 1 a 2 x 2 + x 2 6 a 1 1 a 2 x 2 + C {\displaystyle \int x^{2}\operatorname {arccsc}(a\,x)\,dx={\frac {x^{3}\operatorname {arccsc}(a\,x)}{3}}\,+\,{\frac {1}{6\,a^{3}}}\,\operatorname {arctanh} \,{\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}(a\,x)\,dx={\frac {x^{m+1}\operatorname {arccsc}(a\,x)}{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)}
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