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.

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

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

Obrada nije uspjela. (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int \operatorname {sh} ^{2}cx\,dx={\frac {1}{4c}}\operatorname {sh} 2cx-{\frac {x}{2}}+C}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: $\int \operatorname {sh} ^{n}cx\,dx={\frac {1}{c(n+1)}}\operatorname {sh} ^{n+1}cx\operatorname {ch} cx-{\frac {n+2}{n+1}}\int \operatorname {sh} ^{n+2}cx\,dx+C\qquad {\mbox{(za }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: $\int \operatorname {ch} ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\operatorname {sh} cx\operatorname {ch} ^{n+1}cx-{\frac {n+2}{n+1}}\int \operatorname {ch} ^{n+2}cx\,dx+C\qquad {\mbox{(za }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{\operatorname{sh} cx} = \frac{1}{c} \ln\left|\operatorname{th}\frac{cx}{2}\right| + C}

također: Obrada nije uspjela. (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {\frac {dx}{\operatorname {sh} cx}}={\frac {1}{c}}\ln \left|{\frac {\operatorname {ch} cx-1}{\operatorname {sh} cx}}\right|+C}

također: Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{\operatorname{sh} cx} = \frac{1}{c} \ln\left|\frac{\operatorname{sh} cx}{\operatorname{ch} cx + 1}\right| + C}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dx}{\operatorname{ch} cx} = \frac{2}{c} \operatorname{arctg} e^{cx} + C}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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{)}}

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

također: Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: Obrada nije uspjela. (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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{)}}}

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: $\int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n}cx}}dx={\frac {\operatorname {sh} ^{m+1}cx}{c(n-1)\operatorname {ch} ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\operatorname {sh} ^{m}cx}{\operatorname {ch} ^{n-2}cx}}dx+C\qquad {\mbox{(za }}n\neq 1{\mbox{)}}$

također: Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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{)}}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x\operatorname{ch} cx\,dx = \frac{1}{c} x\operatorname{sh} cx - \frac{1}{c^2}\operatorname{ch} cx + C}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \operatorname{cth} cx\,dx = \frac{1}{c}\ln|\operatorname{sh} cx| + C}

Obrada nije uspjela. (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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{)}}}

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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{)}}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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{)}}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}

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

Obrada nije uspjela. (MathML sa SVG ili PNG za rezervu (preporučljivo za moderne preglednike i alate za pristupačnost): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}

Obrada nije uspjela. (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}