Список интегралов от гиперболических функций

28.01.2021

Список интегралов (первообразных функций) от гиперболических функций. В списке везде опущена аддитивная константа интегрирования.

∫ 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}cxoperatorname {ch} cx-{frac {n-1}{n}}int operatorname {sh} ^{n-2}cx,dxqquad {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}cxoperatorname {ch} cx-{frac {n+2}{n+1}}int operatorname {sh} ^{n+2}cx,dxqquad {mbox{( }}n<0{mbox{, }}n eq -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} cxoperatorname {ch} ^{n-1}cx+{frac {n-1}{n}}int operatorname {ch} ^{n-2}cx,dxqquad {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} cxoperatorname {ch} ^{n+1}cx-{frac {n+2}{n+1}}int operatorname {ch} ^{n+2}cx,dxqquad {mbox{(}}n<0{mbox{, }}n eq -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 ⁡ | 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}} ight|={frac {1}{c}}ln left|{frac {operatorname {ch} cx-1}{operatorname {sh} cx}} ight|={frac {1}{c}}ln left|{frac {operatorname {sh} cx}{operatorname {ch} cx+1}} ight|={frac {1}{c}}ln left|{frac {operatorname {ch} cx-1}{operatorname {ch} cx+1}} ight|} ∫ 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 eq 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 eq 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}}dxqquad {mbox{( }}m eq 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}}dxqquad {mbox{( }}m eq 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}}dxqquad {mbox{( }}m eq 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}}dxqquad {mbox{( }}m eq 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}}dxqquad {mbox{( }}n eq 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}}dxqquad {mbox{( }}n eq 1{mbox{)}}} ∫ x sh ⁡ c x d x = 1 c x ch ⁡ c x − 1 c 2 sh ⁡ c x {displaystyle int xoperatorname {sh} cx,dx={frac {1}{c}}xoperatorname {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 xoperatorname {ch} cx,dx={frac {1}{c}}xoperatorname {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,dxqquad {mbox{( }}n eq 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,dxqquad {mbox{( }}n eq 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} bxoperatorname {sh} cx,dx={frac {1}{b^{2}-c^{2}}}(boperatorname {sh} cxoperatorname {ch} bx-coperatorname {ch} cxoperatorname {sh} bx)qquad {mbox{( }}b^{2} eq 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} bxoperatorname {ch} cx,dx={frac {1}{b^{2}-c^{2}}}(boperatorname {sh} bxoperatorname {ch} cx-coperatorname {sh} cxoperatorname {ch} bx)qquad {mbox{( }}b^{2} eq 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} bxoperatorname {sh} cx,dx={frac {1}{b^{2}-c^{2}}}(boperatorname {sh} bxoperatorname {sh} cx-coperatorname {ch} bxoperatorname {ch} cx)qquad {mbox{( }}b^{2} eq 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)}

Библиография

Книги

• 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
• Корн Г. А., Корн Т. М. Справочник по математике для научных работников и инженеров. — М.: «Наука», 1974.

Таблицы интегралов

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

Вычисление интегралов

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