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Formelsammlung Mathematik: Unbestimmte Integrale hyperbolischer Funktionen
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Formelsammlung Mathematik
Nachfolgende Liste enthält einige Integrale
hyperbolischer Funktionen
.
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∫
sinh
c
x
d
x
=
1
c
cosh
c
x
{\displaystyle \int \sinh cx\,dx={\frac {1}{c}}\cosh cx}
∫
cosh
c
x
d
x
=
1
c
sinh
c
x
{\displaystyle \int \cosh cx\,dx={\frac {1}{c}}\sinh cx}
∫
sinh
2
c
x
d
x
=
1
2
c
sinh
c
x
cosh
c
x
−
x
2
{\displaystyle \int \sinh ^{2}cx\,dx={\frac {1}{2c}}\sinh cx\cosh cx-{\frac {x}{2}}}
∫
cosh
2
c
x
d
x
=
1
2
c
sinh
c
x
cosh
c
x
+
x
2
{\displaystyle \int \cosh ^{2}cx\,dx={\frac {1}{2c}}\sinh cx\cosh cx+{\frac {x}{2}}}
∫
sinh
n
c
x
d
x
=
1
c
n
sinh
n
−
1
c
x
cosh
c
x
−
n
−
1
n
∫
sinh
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}
oder:
∫
sinh
n
c
x
d
x
=
1
c
(
n
+
1
)
sinh
n
+
1
c
x
cosh
c
x
−
n
+
2
n
+
1
∫
sinh
n
+
2
c
x
d
x
(
n
<
0
,
n
≠
−
1
)
{\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}
∫
cosh
n
c
x
d
x
=
1
c
n
sinh
c
x
cosh
n
−
1
c
x
+
n
−
1
n
∫
cosh
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \cosh ^{n}cx\,dx={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}
oder:
∫
cosh
n
c
x
d
x
=
−
1
c
(
n
+
1
)
sinh
c
x
cosh
n
+
1
c
x
+
n
+
2
n
+
1
∫
cosh
n
+
2
c
x
d
x
(
n
<
0
,
n
≠
−
1
)
{\displaystyle \int \cosh ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx+{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}
∫
d
x
sinh
c
x
=
1
c
ln
|
tanh
c
x
2
|
{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|}
oder:
∫
d
x
sinh
c
x
=
1
c
ln
|
cosh
c
x
−
1
sinh
c
x
|
{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|}
oder:
∫
d
x
sinh
c
x
=
1
c
ln
|
sinh
c
x
cosh
c
x
+
1
|
{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|}
oder:
∫
d
x
sinh
c
x
=
1
c
ln
|
cosh
c
x
−
1
cosh
c
x
+
1
|
{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|}
∫
d
x
cosh
c
x
=
2
c
arctan
e
c
x
{\displaystyle \int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}}
∫
d
x
sinh
n
c
x
=
cosh
c
x
c
(
n
−
1
)
sinh
n
−
1
c
x
−
n
−
2
n
−
1
∫
d
x
sinh
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
d
x
cosh
n
c
x
=
sinh
c
x
c
(
n
−
1
)
cosh
n
−
1
c
x
+
n
−
2
n
−
1
∫
d
x
cosh
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
cosh
n
c
x
sinh
m
c
x
d
x
=
cosh
n
−
1
c
x
c
(
n
−
m
)
sinh
m
−
1
c
x
+
n
−
1
n
−
m
∫
cosh
n
−
2
c
x
sinh
m
c
x
d
x
(
m
≠
n
)
{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}
oder:
∫
cosh
n
c
x
sinh
m
c
x
d
x
=
−
cosh
n
+
1
c
x
c
(
m
−
1
)
sinh
m
−
1
c
x
+
n
−
m
+
2
m
−
1
∫
cosh
n
c
x
sinh
m
−
2
c
x
d
x
(
m
≠
1
)
{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}
oder:
∫
cosh
n
c
x
sinh
m
c
x
d
x
=
−
cosh
n
−
1
c
x
c
(
m
−
1
)
sinh
m
−
1
c
x
+
n
−
1
m
−
1
∫
cosh
n
−
2
c
x
sinh
m
−
2
c
x
d
x
(
m
≠
1
)
{\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}
∫
sinh
m
c
x
cosh
n
c
x
d
x
=
sinh
m
−
1
c
x
c
(
m
−
n
)
cosh
n
−
1
c
x
+
m
−
1
m
−
n
∫
sinh
m
−
2
c
x
cosh
n
c
x
d
x
(
m
≠
n
)
{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}
oder:
∫
sinh
m
c
x
cosh
n
c
x
d
x
=
sinh
m
+
1
c
x
c
(
n
−
1
)
cosh
n
−
1
c
x
+
m
−
n
+
2
n
−
1
∫
sinh
m
c
x
cosh
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
oder:
∫
sinh
m
c
x
cosh
n
c
x
d
x
=
−
sinh
m
−
1
c
x
c
(
n
−
1
)
cosh
n
−
1
c
x
+
m
−
1
n
−
1
∫
sinh
m
−
2
c
x
cosh
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
x
sinh
c
x
d
x
=
1
c
x
cosh
c
x
−
1
c
2
sinh
c
x
{\displaystyle \int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx}
∫
x
cosh
c
x
d
x
=
1
c
x
sinh
c
x
−
1
c
2
cosh
c
x
{\displaystyle \int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx}
∫
tanh
c
x
d
x
=
1
c
ln
|
cosh
c
x
|
{\displaystyle \int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|}
∫
coth
c
x
d
x
=
1
c
ln
|
sinh
c
x
|
{\displaystyle \int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|}
∫
tanh
n
c
x
d
x
=
−
1
c
(
n
−
1
)
tanh
n
−
1
c
x
+
∫
tanh
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
coth
n
c
x
d
x
=
−
1
c
(
n
−
1
)
coth
n
−
1
c
x
+
∫
coth
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
sinh
b
x
sinh
c
x
d
x
=
1
b
2
−
c
2
(
b
sinh
c
x
cosh
b
x
−
c
cosh
c
x
sinh
b
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
cosh
b
x
cosh
c
x
d
x
=
1
b
2
−
c
2
(
b
sinh
b
x
cosh
c
x
−
c
sinh
c
x
cosh
b
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
cosh
b
x
sinh
c
x
d
x
=
1
b
2
−
c
2
(
b
sinh
b
x
sinh
c
x
−
c
cosh
b
x
cosh
c
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)}
∫
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)}
∫
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)}
∫
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)}
Siehe auch
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