Giperbolik funksiyalarning integrallar jadvali
Izoh. Hamma integrallarda oʻzgarmas qoʻshiluvchi tushirib qoldirilgan.
∫
sh
c
x
d
x
=
1
c
ch
c
x
{\displaystyle \int \operatorname {sh} cx\,dx={\frac {1}{c}}\operatorname {ch} cx}
∫
ch
c
x
d
x
=
1
c
sh
c
x
{\displaystyle \int \operatorname {ch} cx\,dx={\frac {1}{c}}\operatorname {sh} cx}
∫
sh
2
c
x
d
x
=
1
4
c
sh
2
c
x
−
x
2
{\displaystyle \int \operatorname {sh} ^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh} 2cx-{\frac {x}{2}}}
∫
ch
2
c
x
d
x
=
1
4
c
sh
2
c
x
+
x
2
{\displaystyle \int \operatorname {ch} ^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh} 2cx+{\frac {x}{2}}}
∫
sh
n
c
x
d
x
=
1
c
n
sh
n
−
1
c
x
ch
c
x
−
n
−
1
n
∫
sh
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{cn}}\operatorname {sh} ^{n-1}cx\operatorname {ch} cx-{\frac {n-1}{n}}\int \operatorname {sh} ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}
также:
∫
sh
n
c
x
d
x
=
1
c
(
n
+
1
)
sh
n
+
1
c
x
ch
c
x
−
n
+
2
n
+
1
∫
sh
n
+
2
c
x
d
x
(
n
<
0
,
n
≠
−
1
)
{\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{c(n+1)}}\operatorname {sh} ^{n+1}cx\operatorname {ch} cx-{\frac {n+2}{n+1}}\int \operatorname {sh} ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}
∫
ch
n
c
x
d
x
=
1
c
n
sh
c
x
ch
n
−
1
c
x
+
n
−
1
n
∫
ch
n
−
2
c
x
d
x
(
n
>
0
)
{\displaystyle \int \operatorname {ch} ^{n}cx\,dx={\frac {1}{cn}}\operatorname {sh} cx\operatorname {ch} ^{n-1}cx+{\frac {n-1}{n}}\int \operatorname {ch} ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}}
также:
∫
ch
n
c
x
d
x
=
−
1
c
(
n
+
1
)
sh
c
x
ch
n
+
1
c
x
−
n
+
2
n
+
1
∫
ch
n
+
2
c
x
d
x
(
n
<
0
,
n
≠
−
1
)
{\displaystyle \int \operatorname {ch} ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\operatorname {sh} cx\operatorname {ch} ^{n+1}cx-{\frac {n+2}{n+1}}\int \operatorname {ch} ^{n+2}cx\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}}
∫
d
x
sh
c
x
=
1
c
ln
|
th
c
x
2
|
=
1
c
ln
|
ch
c
x
−
1
sh
c
x
|
=
1
c
ln
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sh
c
x
ch
c
x
+
1
|
=
1
c
ln
|
ch
c
x
−
1
ch
c
x
+
1
|
{\displaystyle \int {\frac {dx}{\operatorname {sh} cx}}={\frac {1}{c}}\ln \left|\operatorname {th} {\frac {cx}{2}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {sh} cx}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {sh} cx}{\operatorname {ch} cx+1}}\right|={\frac {1}{c}}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {ch} cx+1}}\right|}
∫
d
x
sh
2
c
x
=
−
1
c
cth
c
x
{\displaystyle \int {\frac {dx}{\operatorname {sh} ^{2}cx}}=-{\frac {1}{c}}\operatorname {cth} cx}
∫
d
x
ch
c
x
=
2
c
arctg
e
c
x
{\displaystyle \int {\frac {dx}{\operatorname {ch} cx}}={\frac {2}{c}}\operatorname {arctg} e^{cx}}
∫
d
x
ch
2
c
x
=
1
c
th
c
x
{\displaystyle \int {\frac {dx}{\operatorname {ch} ^{2}cx}}={\frac {1}{c}}\operatorname {th} cx}
∫
d
x
sh
n
c
x
=
ch
c
x
c
(
n
−
1
)
sh
n
−
1
c
x
−
n
−
2
n
−
1
∫
d
x
sh
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\operatorname {sh} ^{n}cx}}={\frac {\operatorname {ch} cx}{c(n-1)\operatorname {sh} ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\operatorname {sh} ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
d
x
ch
n
c
x
=
sh
c
x
c
(
n
−
1
)
ch
n
−
1
c
x
+
n
−
2
n
−
1
∫
d
x
ch
n
−
2
c
x
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{\operatorname {ch} ^{n}cx}}={\frac {\operatorname {sh} cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\operatorname {ch} ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
ch
n
c
x
sh
m
c
x
d
x
=
ch
n
−
1
c
x
c
(
n
−
m
)
sh
m
−
1
c
x
+
n
−
1
n
−
m
∫
ch
n
−
2
c
x
sh
m
c
x
d
x
(
m
≠
n
)
{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx={\frac {\operatorname {ch} ^{n-1}cx}{c(n-m)\operatorname {sh} ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\operatorname {ch} ^{n-2}cx}{\operatorname {sh} ^{m}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}
также:
∫
ch
n
c
x
sh
m
c
x
d
x
=
−
ch
n
+
1
c
x
c
(
m
−
1
)
sh
m
−
1
c
x
+
n
−
m
+
2
m
−
1
∫
ch
n
c
x
sh
m
−
2
c
x
d
x
(
m
≠
1
)
{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx=-{\frac {\operatorname {ch} ^{n+1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}
также:
∫
ch
n
c
x
sh
m
c
x
d
x
=
−
ch
n
−
1
c
x
c
(
m
−
1
)
sh
m
−
1
c
x
+
n
−
1
m
−
1
∫
ch
n
−
2
c
x
sh
m
−
2
c
x
d
x
(
m
≠
1
)
{\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}}dx=-{\frac {\operatorname {ch} ^{n-1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\operatorname {ch} ^{n-2}cx}{\operatorname {sh} ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}}
∫
sh
m
c
x
ch
n
c
x
d
x
=
sh
m
−
1
c
x
c
(
m
−
n
)
ch
n
−
1
c
x
+
m
−
1
m
−
n
∫
sh
m
−
2
c
x
ch
n
c
x
d
x
(
m
≠
n
)
{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx={\frac {\operatorname {sh} ^{m-1}cx}{c(m-n)\operatorname {ch} ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\operatorname {sh} ^{m-2}cx}{\operatorname {ch} ^{n}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}}
также:
∫
sh
m
c
x
ch
n
c
x
d
x
=
sh
m
+
1
c
x
c
(
n
−
1
)
ch
n
−
1
c
x
+
m
−
n
+
2
n
−
1
∫
sh
m
c
x
ch
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx={\frac {\operatorname {sh} ^{m+1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
также:
∫
sh
m
c
x
ch
n
c
x
d
x
=
−
sh
m
−
1
c
x
c
(
n
−
1
)
ch
n
−
1
c
x
+
m
−
1
n
−
1
∫
sh
m
−
2
c
x
ch
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx=-{\frac {\operatorname {sh} ^{m-1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\operatorname {sh} ^{m-2}cx}{\operatorname {ch} ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
x
sh
c
x
d
x
=
1
c
x
ch
c
x
−
1
c
2
sh
c
x
{\displaystyle \int x\operatorname {sh} cx\,dx={\frac {1}{c}}x\operatorname {ch} cx-{\frac {1}{c^{2}}}\operatorname {sh} cx}
∫
x
ch
c
x
d
x
=
1
c
x
sh
c
x
−
1
c
2
ch
c
x
{\displaystyle \int x\operatorname {ch} cx\,dx={\frac {1}{c}}x\operatorname {sh} cx-{\frac {1}{c^{2}}}\operatorname {ch} cx}
∫
th
c
x
d
x
=
1
c
ln
|
ch
c
x
|
{\displaystyle \int \operatorname {th} cx\,dx={\frac {1}{c}}\ln |\operatorname {ch} cx|}
∫
cth
c
x
d
x
=
1
c
ln
|
sh
c
x
|
{\displaystyle \int \operatorname {cth} cx\,dx={\frac {1}{c}}\ln |\operatorname {sh} cx|}
∫
th
2
c
x
d
x
=
x
−
1
c
th
c
x
{\displaystyle \int \operatorname {th} ^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {th} cx}
∫
cth
2
c
x
d
x
=
x
−
1
c
cth
c
x
{\displaystyle \int \operatorname {cth} ^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {cth} cx}
∫
th
n
c
x
d
x
=
−
1
c
(
n
−
1
)
th
n
−
1
c
x
+
∫
th
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \operatorname {th} ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\operatorname {th} ^{n-1}cx+\int \operatorname {th} ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{ )}}}
∫
cth
n
c
x
d
x
=
−
1
c
(
n
−
1
)
cth
n
−
1
c
x
+
∫
cth
n
−
2
c
x
d
x
(
n
≠
1
)
{\displaystyle \int \operatorname {cth} ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\operatorname {cth} ^{n-1}cx+\int \operatorname {cth} ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
∫
sh
b
x
sh
c
x
d
x
=
1
b
2
−
c
2
(
b
sh
c
x
ch
b
x
−
c
ch
c
x
sh
b
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \operatorname {sh} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} cx\operatorname {ch} bx-c\operatorname {ch} cx\operatorname {sh} bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
ch
b
x
ch
c
x
d
x
=
1
b
2
−
c
2
(
b
sh
b
x
ch
c
x
−
c
sh
c
x
ch
b
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \operatorname {ch} bx\operatorname {ch} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} bx\operatorname {ch} cx-c\operatorname {sh} cx\operatorname {ch} bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
ch
b
x
sh
c
x
d
x
=
1
b
2
−
c
2
(
b
sh
b
x
sh
c
x
−
c
ch
b
x
ch
c
x
)
(
b
2
≠
c
2
)
{\displaystyle \int \operatorname {ch} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\operatorname {sh} bx\operatorname {sh} cx-c\operatorname {ch} bx\operatorname {ch} cx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}}
∫
sh
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
ch
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
sh
(
a
x
+
b
)
cos
(
c
x
+
d
)
{\displaystyle \int \operatorname {sh} (ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\operatorname {ch} (ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\operatorname {sh} (ax+b)\cos(cx+d)}
∫
sh
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
ch
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
sh
(
a
x
+
b
)
sin
(
c
x
+
d
)
{\displaystyle \int \operatorname {sh} (ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\operatorname {ch} (ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\operatorname {sh} (ax+b)\sin(cx+d)}
∫
ch
(
a
x
+
b
)
sin
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a
2
+
c
2
ch
(
a
x
+
b
)
cos
(
c
x
+
d
)
{\displaystyle \int \operatorname {ch} (ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\operatorname {sh} (ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\operatorname {ch} (ax+b)\cos(cx+d)}
∫
ch
(
a
x
+
b
)
cos
(
c
x
+
d
)
d
x
=
a
a
2
+
c
2
sh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a
2
+
c
2
ch
(
a
x
+
b
)
sin
(
c
x
+
d
)
{\displaystyle \int \operatorname {ch} (ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\operatorname {sh} (ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\operatorname {ch} (ax+b)\sin(cx+d)}
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