Гиперболăлла функцисен интегралĕсен йышĕ

Гиперболăлла функцисен интегралĕсен йышĕ. Аяларах çавнашкал интегралсен (умсăнарсен) йышне илсе кăтартнă. Пур çĕрте те аддитивлă констаттăна катертнĕ.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Библиографи

Кĕнекесем

• Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
• Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
• D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4

Интегралсен таблицисем

• Интегралы на EqWorld
• S.O.S. Mathematics: Tables and Formulas

Интегралсене шутлани

• The Integrator (Wolfram Research)
• Империя Чисел
• Методы вычисления неопределённых интегралов