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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