Seznam integrálů hyperbolických funkcí

Seznam integrálů ( primitivních ) hyperbolických funkcí . Aditivní konstanta integrace je v seznamu všude vynechána .

\int \operatorname {sh}cx\,dx={\frac {1}{c}}\operatorname {ch}cx

\int \operatorname {ch}cx\,dx={\frac {1}{c}}\operatorname {sh}cx

\int \operatorname {sh}^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh}2cx-{\frac {x}{2}}

\int \operatorname {ch}^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh}2cx+{\frac {x}{2}}

\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{)))

taky:

\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{)))

\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{)))

taky:

\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{)))

\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|

\int {\frac {dx}{\operatorname {sh}^{2}cx}}=-{\frac {1}{c}}\operatorname {cth}cx

\int {\frac {dx}{\operatorname {ch}cx}}={\frac {2}{c}}\operatorname {arctg}e^{{cx}}

\int {\frac {dx}{\operatorname {ch}^{2}cx}}={\frac {1}{c}}\operatorname {th}cx

\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{)))

\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{)))

\int {\frac {\operatorname {ch}^{n}cx}{\operatorname {sh}^{m}cx}}dx={\frac {\operatorname {ch}^{{n-1}}cx }{c(nm)\operatorname {sh}^{{m-1}}cx}}+{\frac {n-1}{nm}}\int {\frac {\operatorname {ch}^{{n -2}}cx}{\operatorname {sh}^{m}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)))

taky:

\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{)))

taky:

\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}{\název operátora {sh}^{{m-2}}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)))

\int {\frac {\operatorname {sh}^{m}cx}{\operatorname {ch}^{n}cx}}dx={\frac {\operatorname {sh}^{{m-1}}cx }{c(mn)\operatorname {ch}^{{n-1}}cx}}+{\frac {m-1}{mn}}\int {\frac {\operatorname {sh}^{{m -2}}cx}{\operatorname {ch}^{n}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)))

taky:

\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}{\název operátora {ch}^{{n-2}}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)))

taky:

\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}{\název operátora {ch}^{{n-2}}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)))

\int x\operatorname {sh}cx\,dx={\frac {1}{c}}x\operatorname {ch}cx-{\frac {1}{c^{2}}}\operatorname {sh} cx

\int x\operatorname {ch}cx\,dx={\frac {1}{c}}x\operatorname {sh}cx-{\frac {1}{c^{2}}}\operatorname {ch} cx

\int \operatorname {th}cx\,dx={\frac {1}{c}}\ln |\operatorname {ch}cx|

\int \operatorname {cth}cx\,dx={\frac {1}{c}}\ln |\operatorname {sh}cx|

\int \operatorname {th}^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {th}cx

\int \operatorname {cth}^{2}cx\,dx=x-{\frac {1}{c}}\operatorname {cth}cx

\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{ )))

\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{)))

\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{)))

\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{)))

\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{)))

\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}}}\jméno operátora {sh}(ax+b)\cos(cx+d)

\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))}\jméno operátora {sh}(ax+b)\sin(cx+d)

\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))}\jméno operátora {ch}(ax+b)\cos(cx+d)

\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}}}\jméno operátora {ch}(ax+b)\sin(cx+d)

Bibliografie

knihy

Gradshtein I. S., Ryzhik I. M. Tabulky integrálů, součtů, řad a součinů. - 4. vyd. - M .: Nauka, 1963. - ISBN 0-12-294757-6 // EqWorld
Dvait G. B. Tabulky integrálů Petrohrad: Nakladatelství a tiskárna JSC VNIIG im. B. V. Vedeneeva, 1995. - 176 s. — ISBN 5-85529-029-8 . // EqWorld
D. Zwillinger. CRC Standardní matematické tabulky a vzorce , 31. vydání, 2002. ISBN 1-58488-291-3 .
M. Abramowitz a I. A. Stegun, ed. Příručka matematických funkcí se vzorci, grafy a matematickými tabulkami , 1964. ISBN 0-486-61272-4
Korn G. A., Korn T. M. Příručka matematiky pro vědce a inženýry . - M .: " Nauka ", 1974.

Tabulky integrálů

Výpočet integrálů

Seznamy integrálů podle typů funkcí
Základní Racionální iracionální trigonometrický zvrátit hyperbolický zvrátit exponenciální logaritmické funkce