Popis integrala hiperbolnih funkcija

Slijedi popis integrala (antiderivacija funkcija) hiperbolnih funkcija. Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala.

Za konstantu c se pretpostavlja da je različita od nule.

${\displaystyle \int \operatorname {sh} cx\,dx={\frac {1}{c}\operatorname {ch} cx+C}$

${\displaystyle \int \operatorname {ch} cx\,dx={\frac {1}{c}\operatorname {sh} cx+C}$

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

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

${\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{cn}\operatorname {sh} ^{n-1}cx\operatorname {ch} cx-{\frac {n-1}{n}\int \operatorname {sh} ^{n-2}cx\,dx+C\qquad {\mbox{(za }n>0{\mbox{)}$

također: ${\displaystyle \int \operatorname {sh} ^{n}cx\,dx={\frac {1}{c(n+1)}\operatorname {sh} ^{n+1}cx\operatorname {ch} cx-{\frac {n+2}{n+1}\int \operatorname {sh} ^{n+2}cx\,dx+C\qquad {\mbox{(za }n<0{\mbox{, }n\neq -1{\mbox{)}$

${\displaystyle \int \operatorname {ch} ^{n}cx\,dx={\frac {1}{cn}\operatorname {sh} cx\operatorname {ch} ^{n-1}cx+{\frac {n-1}{n}\int \operatorname {ch} ^{n-2}cx\,dx+C\qquad {\mbox{(za }n>0{\mbox{)}$

također: ${\displaystyle \int \operatorname {ch} ^{n}cx\,dx=-{\frac {1}{c(n+1)}\operatorname {sh} cx\operatorname {ch} ^{n+1}cx-{\frac {n+2}{n+1}\int \operatorname {ch} ^{n+2}cx\,dx+C\qquad {\mbox{(za }n<0{\mbox{, }n\neq -1{\mbox{)}$

${\displaystyle \int {\frac {dx}{\operatorname {sh} cx}={\frac {1}{c}\ln \left|\operatorname {th} {\frac {cx}{2}\right|+C}$

također: ${\displaystyle \int {\frac {dx}{\operatorname {sh} cx}={\frac {1}{c}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {sh} cx}\right|+C}$

također: ${\displaystyle \int {\frac {dx}{\operatorname {sh} cx}={\frac {1}{c}\ln \left|{\frac {\operatorname {sh} cx}{\operatorname {ch} cx+1}\right|+C}$

također: ${\displaystyle \int {\frac {dx}{\operatorname {sh} cx}={\frac {1}{c}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {ch} cx+1}\right|+C}$

${\displaystyle \int {\frac {dx}{\operatorname {ch} cx}={\frac {2}{c}\operatorname {arctg} e^{cx}+C}$

${\displaystyle \int {\frac {dx}{\operatorname {sh} ^{n}cx}={\frac {\operatorname {ch} cx}{c(n-1)\operatorname {sh} ^{n-1}cx}-{\frac {n-2}{n-1}\int {\frac {dx}{\operatorname {sh} ^{n-2}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

${\displaystyle \int {\frac {dx}{\operatorname {ch} ^{n}cx}={\frac {\operatorname {sh} cx}{c(n-1)\operatorname {ch} ^{n-1}cx}+{\frac {n-2}{n-1}\int {\frac {dx}{\operatorname {ch} ^{n-2}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

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

također: ${\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}dx=-{\frac {\operatorname {ch} ^{n+1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}+{\frac {n-m+2}{m-1}\int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m-2}cx}dx+C\qquad {\mbox{(za }m\neq 1{\mbox{)}$

također: ${\displaystyle \int {\frac {\operatorname {ch} ^{n}cx}{\operatorname {sh} ^{m}cx}dx=-{\frac {\operatorname {ch} ^{n-1}cx}{c(m-1)\operatorname {sh} ^{m-1}cx}+{\frac {n-1}{m-1}\int {\frac {\operatorname {ch} ^{n-2}cx}{\operatorname {sh} ^{m-2}cx}dx+C\qquad {\mbox{(za }m\neq 1{\mbox{)}$

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

također: ${\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}dx={\frac {\operatorname {sh} ^{m+1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}+{\frac {m-n+2}{n-1}\int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n-2}cx}dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

također: ${\displaystyle \int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}dx=-{\frac {\operatorname {sh} ^{m-1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}+{\frac {m-1}{n-1}\int {\frac {\operatorname {sh} ^{m-2}cx}{\operatorname {ch} ^{n-2}cx}dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

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

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

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

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

${\displaystyle \int \operatorname {th} ^{n}cx\,dx=-{\frac {1}{c(n-1)}\operatorname {th} ^{n-1}cx+\int \operatorname {th} ^{n-2}cx\,dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

${\displaystyle \int \operatorname {cth} ^{n}cx\,dx=-{\frac {1}{c(n-1)}\operatorname {cth} ^{n-1}cx+\int \operatorname {cth} ^{n-2}cx\,dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

${\displaystyle \int \operatorname {sh} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}(b\operatorname {sh} cx\operatorname {ch} bx-c\operatorname {ch} cx\operatorname {sh} bx)+C\qquad {\mbox{(za }b^{2}\neq c^{2}{\mbox{)}$

${\displaystyle \int \operatorname {ch} bx\operatorname {ch} cx\,dx={\frac {1}{b^{2}-c^{2}(b\operatorname {sh} bx\operatorname {ch} cx-c\operatorname {sh} cx\operatorname {ch} bx)+C\qquad {\mbox{(za }b^{2}\neq c^{2}{\mbox{)}$

${\displaystyle \int \operatorname {ch} bx\operatorname {sh} cx\,dx={\frac {1}{b^{2}-c^{2}(b\operatorname {sh} bx\operatorname {sh} cx-c\operatorname {ch} bx\operatorname {ch} cx)+C\qquad {\mbox{(za }b^{2}\neq c^{2}{\mbox{)}$

${\displaystyle \int \operatorname {sh} (ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}\operatorname {ch} (ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}\operatorname {sh} (ax+b)\cos(cx+d)+C}$

${\displaystyle \int \operatorname {sh} (ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}\operatorname {ch} (ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}\operatorname {sh} (ax+b)\sin(cx+d)+C}$

${\displaystyle \int \operatorname {ch} (ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}\operatorname {sh} (ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}\operatorname {ch} (ax+b)\cos(cx+d)+C}$

${\displaystyle \int \operatorname {ch} (ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}\operatorname {sh} (ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}\operatorname {ch} (ax+b)\sin(cx+d)+C}$