از ویکی‌پدیا، دانشنامهٔ آزاد
در ادامه فهرستی از انتگرال تابع‌های وارون هذلولوی نوشته شده‌است. برای آگاهی بیشتر صفحهٔ فهرست انتگرال‌ها را نگاه کنید.
در تمامی رابطه‌ها فرض می‌شود که a ناصفر و C ثابت انتگرال‌گیری است. همچنین یادآوری می‌شود که برای هریک از انتگرال‌های وارون هذلولوی، که در زیر نوشته شده‌اند، در مقابل، یک رابطه نیز در فهرست انتگرال تابع‌های وارون مثلثاتی وجود دارد.
[ ویرایش ]
∫
arsinh
(
a
x
)
d
x
=
x
arsinh
(
a
x
)
−
a
2
x
2
+
1
a
+
C
{\displaystyle \int \operatorname {arsinh} (a\,x)\,dx=x\,\operatorname {arsinh} (a\,x)-{\frac {\sqrt {a^{2}\,x^{2}+1}}{a}}+C}
∫
x
arsinh
(
a
x
)
d
x
=
x
2
arsinh
(
a
x
)
2
+
arsinh
(
a
x
)
4
a
2
−
x
a
2
x
2
+
1
4
a
+
C
{\displaystyle \int x\,\operatorname {arsinh} (a\,x)dx={\frac {x^{2}\,\operatorname {arsinh} (a\,x)}{2}}+{\frac {\operatorname {arsinh} (a\,x)}{4\,a^{2}}}-{\frac {x{\sqrt {a^{2}\,x^{2}+1}}}{4\,a}}+C}
∫
x
2
arsinh
(
a
x
)
d
x
=
x
3
arsinh
(
a
x
)
3
−
(
a
2
x
2
−
2
)
a
2
x
2
+
1
9
a
3
+
C
{\displaystyle \int x^{2}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{3}\,\operatorname {arsinh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}-2\right){\sqrt {a^{2}\,x^{2}+1}}}{9\,a^{3}}}+C}
∫
x
m
arsinh
(
a
x
)
d
x
=
x
m
+
1
arsinh
(
a
x
)
m
+
1
−
a
m
+
1
∫
x
m
+
1
a
2
x
2
+
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsinh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {a^{2}\,x^{2}+1}}}\,dx\quad (m\neq -1)}
∫
arsinh
(
a
x
)
2
d
x
=
2
x
+
x
arsinh
(
a
x
)
2
−
2
a
2
x
2
+
1
arsinh
(
a
x
)
a
+
C
{\displaystyle \int \operatorname {arsinh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arsinh} (a\,x)^{2}-{\frac {2\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)}{a}}+C}
∫
arsinh
(
a
x
)
n
d
x
=
x
arsinh
(
a
x
)
n
−
n
a
2
x
2
+
1
arsinh
(
a
x
)
n
−
1
a
+
n
(
n
−
1
)
∫
arsinh
(
a
x
)
n
−
2
d
x
{\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=x\,\operatorname {arsinh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arsinh} (a\,x)^{n-2}\,dx}
∫
arsinh
(
a
x
)
n
d
x
=
−
x
arsinh
(
a
x
)
n
+
2
(
n
+
1
)
(
n
+
2
)
+
a
2
x
2
+
1
arsinh
(
a
x
)
n
+
1
a
(
n
+
1
)
+
1
(
n
+
1
)
(
n
+
2
)
∫
arsinh
(
a
x
)
n
+
2
d
x
(
n
≠
−
1
,
−
2
)
{\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arsinh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n+1}}{a(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arsinh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}
[ ویرایش ]
∫
arcosh
(
a
x
)
d
x
=
x
arcosh
(
a
x
)
−
a
x
+
1
a
x
−
1
a
+
C
{\displaystyle \int \operatorname {arcosh} (a\,x)\,dx=x\,\operatorname {arcosh} (a\,x)-{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{a}}+C}
∫
x
arcosh
(
a
x
)
d
x
=
x
2
arcosh
(
a
x
)
2
−
arcosh
(
a
x
)
4
a
2
−
x
a
x
+
1
a
x
−
1
4
a
+
C
{\displaystyle \int x\,\operatorname {arcosh} (a\,x)dx={\frac {x^{2}\,\operatorname {arcosh} (a\,x)}{2}}-{\frac {\operatorname {arcosh} (a\,x)}{4\,a^{2}}}-{\frac {x\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{4\,a}}+C}
∫
x
2
arcosh
(
a
x
)
d
x
=
x
3
arcosh
(
a
x
)
3
−
(
a
2
x
2
+
2
)
a
x
+
1
a
x
−
1
9
a
3
+
C
{\displaystyle \int x^{2}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{3}\,\operatorname {arcosh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{9\,a^{3}}}+C}
∫
x
m
arcosh
(
a
x
)
d
x
=
x
m
+
1
arcosh
(
a
x
)
m
+
1
−
a
m
+
1
∫
x
m
+
1
a
x
+
1
a
x
−
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arcosh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}}\,dx\quad (m\neq -1)}
∫
arcosh
(
a
x
)
2
d
x
=
2
x
+
x
arcosh
(
a
x
)
2
−
2
a
x
+
1
a
x
−
1
arcosh
(
a
x
)
a
+
C
{\displaystyle \int \operatorname {arcosh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arcosh} (a\,x)^{2}-{\frac {2\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)}{a}}+C}
∫
arcosh
(
a
x
)
n
d
x
=
x
arcosh
(
a
x
)
n
−
n
a
x
+
1
a
x
−
1
arcosh
(
a
x
)
n
−
1
a
+
n
(
n
−
1
)
∫
arcosh
(
a
x
)
n
−
2
d
x
{\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=x\,\operatorname {arcosh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arcosh} (a\,x)^{n-2}\,dx}
∫
arcosh
(
a
x
)
n
d
x
=
−
x
arcosh
(
a
x
)
n
+
2
(
n
+
1
)
(
n
+
2
)
+
a
x
+
1
a
x
−
1
arcosh
(
a
x
)
n
+
1
a
(
n
+
1
)
+
1
(
n
+
1
)
(
n
+
2
)
∫
arcosh
(
a
x
)
n
+
2
d
x
(
n
≠
−
1
,
−
2
)
{\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arcosh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n+1}}{a\,(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arcosh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)}
[ ویرایش ]
∫
artanh
(
a
x
)
d
x
=
x
artanh
(
a
x
)
+
ln
(
a
2
x
2
−
1
)
2
a
+
C
{\displaystyle \int \operatorname {artanh} (a\,x)\,dx=x\,\operatorname {artanh} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+C}
∫
x
artanh
(
a
x
)
d
x
=
x
2
artanh
(
a
x
)
2
−
artanh
(
a
x
)
2
a
2
+
x
2
a
+
C
{\displaystyle \int x\,\operatorname {artanh} (a\,x)dx={\frac {x^{2}\,\operatorname {artanh} (a\,x)}{2}}-{\frac {\operatorname {artanh} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C}
∫
x
2
artanh
(
a
x
)
d
x
=
x
3
artanh
(
a
x
)
3
+
ln
(
a
2
x
2
−
1
)
6
a
3
+
x
2
6
a
+
C
{\displaystyle \int x^{2}\,\operatorname {artanh} (a\,x)dx={\frac {x^{3}\,\operatorname {artanh} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C}
∫
x
m
artanh
(
a
x
)
d
x
=
x
m
+
1
artanh
(
a
x
)
m
+
1
+
a
m
+
1
∫
x
m
+
1
a
2
x
2
−
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {artanh} (a\,x)dx={\frac {x^{m+1}\operatorname {artanh} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)}
[ ویرایش ]
∫
arcoth
(
a
x
)
d
x
=
x
arcoth
(
a
x
)
+
ln
(
a
2
x
2
−
1
)
2
a
+
C
{\displaystyle \int \operatorname {arcoth} (a\,x)\,dx=x\,\operatorname {arcoth} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+C}
∫
x
arcoth
(
a
x
)
d
x
=
x
2
arcoth
(
a
x
)
2
−
arcoth
(
a
x
)
2
a
2
+
x
2
a
+
C
{\displaystyle \int x\,\operatorname {arcoth} (a\,x)dx={\frac {x^{2}\,\operatorname {arcoth} (a\,x)}{2}}-{\frac {\operatorname {arcoth} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C}
∫
x
2
arcoth
(
a
x
)
d
x
=
x
3
arcoth
(
a
x
)
3
+
ln
(
a
2
x
2
−
1
)
6
a
3
+
x
2
6
a
+
C
{\displaystyle \int x^{2}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{3}\,\operatorname {arcoth} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C}
∫
x
m
arcoth
(
a
x
)
d
x
=
x
m
+
1
arcoth
(
a
x
)
m
+
1
+
a
m
+
1
∫
x
m
+
1
a
2
x
2
−
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{m+1}\operatorname {arcoth} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)}
[ ویرایش ]
∫
arsech
(
a
x
)
d
x
=
x
arsech
(
a
x
)
−
2
a
arctan
1
−
a
x
1
+
a
x
+
C
{\displaystyle \int \operatorname {arsech} (a\,x)\,dx=x\,\operatorname {arsech} (a\,x)-{\frac {2}{a}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}+C}
∫
x
arsech
(
a
x
)
d
x
=
x
2
arsech
(
a
x
)
2
−
(
1
+
a
x
)
2
a
2
1
−
a
x
1
+
a
x
+
C
{\displaystyle \int x\,\operatorname {arsech} (a\,x)dx={\frac {x^{2}\,\operatorname {arsech} (a\,x)}{2}}-{\frac {(1+a\,x)}{2\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}+C}
∫
x
2
arsech
(
a
x
)
d
x
=
x
3
arsech
(
a
x
)
3
−
1
3
a
3
arctan
1
−
a
x
1
+
a
x
−
x
(
1
+
a
x
)
6
a
2
1
−
a
x
1
+
a
x
+
C
{\displaystyle \int x^{2}\,\operatorname {arsech} (a\,x)dx={\frac {x^{3}\,\operatorname {arsech} (a\,x)}{3}}\,-\,{\frac {1}{3\,a^{3}}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}\,-\,{\frac {x(1+a\,x)}{6\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}\,+\,C}
∫
x
m
arsech
(
a
x
)
d
x
=
x
m
+
1
arsech
(
a
x
)
m
+
1
+
1
m
+
1
∫
x
m
(
1
+
a
x
)
1
−
a
x
1
+
a
x
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {arsech} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsech} (a\,x)}{m+1}}\,+\,{\frac {1}{m+1}}\int {\frac {x^{m}}{(1+a\,x){\sqrt {\frac {1-a\,x}{1+a\,x}}}}}\,dx\quad (m\neq -1)}
[ ویرایش ]
∫
arcsch
(
a
x
)
d
x
=
x
arcsch
(
a
x
)
+
1
a
artanh
1
a
2
x
2
+
1
+
C
{\displaystyle \int \operatorname {arcsch} (a\,x)\,dx=x\,\operatorname {arcsch} (a\,x)+{\frac {1}{a}}\,\operatorname {artanh} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C}
∫
x
arcsch
(
a
x
)
d
x
=
x
2
arcsch
(
a
x
)
2
+
x
2
a
1
a
2
x
2
+
1
+
C
{\displaystyle \int x\,\operatorname {arcsch} (a\,x)dx={\frac {x^{2}\,\operatorname {arcsch} (a\,x)}{2}}+{\frac {x}{2\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C}
∫
x
2
arcsch
(
a
x
)
d
x
=
x
3
arcsch
(
a
x
)
3
−
1
6
a
3
artanh
1
a
2
x
2
+
1
+
x
2
6
a
1
a
2
x
2
+
1
+
C
{\displaystyle \int x^{2}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{3}\,\operatorname {arcsch} (a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {artanh} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,C}
∫
x
m
arcsch
(
a
x
)
d
x
=
x
m
+
1
arcsch
(
a
x
)
m
+
1
+
1
a
(
m
+
1
)
∫
x
m
−
1
1
a
2
x
2
+
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{m+1}\operatorname {arcsch} (a\,x)}{m+1}}\,+\,{\frac {1}{a(m+1)}}\int {\frac {x^{m-1}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}}\,dx\quad (m\neq -1)}
مشارکت‌کنندگان ویکی‌پدیا. «List of integrals of inverse hyperbolic functions دانشنامهٔ ویکی‌پدیای انگلیسی
