Tabelul integralelor nedeterminate ale funcțiilor hiperbolice

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Această pagină conține un tabel de integrale nedeterminate ale funcțiilor hiperbolice . Pentru alte integrale a se vedea Integral § Tabelele de integrale .

\int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx

{\ displaystyle \ int \ sinh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ cosh cx}

{\ displaystyle \ int \ sinh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ cosh cx}

\int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx

{\ displaystyle \ int \ cosh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ sinh cx}

{\ displaystyle \ int \ cosh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ sinh cx}

\int \sinh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}

{\ displaystyle \ int \ sinh ^ {2} cx \, \ mathrm {d} x = {\ frac {1} {4c}} \ sinh 2cx - {\ frac {x} {2}}}

{\ displaystyle \ int \ sinh ^ {2} cx \, \ mathrm {d} x = {\ frac {1} {4c}} \ sinh 2cx - {\ frac {x} {2}}}

\int \cosh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}

{\ displaystyle \ int \ cosh ^ {2} cx \, \ mathrm {d} x = {\ frac {1} {4c}} \ sinh 2cx + {\ frac {x} {2}}}

{\ displaystyle \ int \ cosh ^ {2} cx \, \ mathrm {d} x = {\ frac {1} {4c}} \ sinh 2cx + {\ frac {x} {2}}}

\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n>0{\mbox{)}}

{\ displaystyle \ int \ sinh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {cn}} \ sinh ^ {n-1} cx \ cosh cx - {\ frac {n- 1} {n}} \ int \ sinh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n> 0 {\ mbox {)}}}

{\ displaystyle \ int \ sinh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {cn}} \ sinh ^ {n-1} cx \ cosh cx - {\ frac {n- 1} {n}} \ int \ sinh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n> 0 {\ mbox {)}}}

de asemenea:

\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}

{\ displaystyle \ int \ sinh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {c (n + 1)}} \ sinh ^ {n + 1} cx \ cosh cx- { \ frac {n + 2} {n + 1}} \ int \ sinh ^ {n + 2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n <0 {\ mbox {, }} n \ neq -1 {\ mbox {)}}}

{\ displaystyle \ int \ sinh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {c (n + 1)}} \ sinh ^ {n + 1} cx \ cosh cx- { \ frac {n + 2} {n + 1}} \ int \ sinh ^ {n + 2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n <0 {\ mbox {, }} n \ neq -1 {\ mbox {)}}}

\int \cosh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n>0{\mbox{)}}

{\ displaystyle \ int \ cosh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {cn}} \ sinh cx \ cosh ^ {n-1} cx + {\ frac {n- 1} {n}} \ int \ cosh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n> 0 {\ mbox {)}}}

{\ displaystyle \ int \ cosh ^ {n} cx \, \ mathrm {d} x = {\ frac {1} {cn}} \ sinh cx \ cosh ^ {n-1} cx + {\ frac {n- 1} {n}} \ int \ cosh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n> 0 {\ mbox {)}}}

de asemenea:

\int \cosh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}

{\ displaystyle \ int \ cosh ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n + 1)}} \ sinh cx \ cosh ^ {n + 1} cx- {\ frac {n + 2} {n + 1}} \ int \ cosh ^ {n + 2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n <0 {\ mbox { ,}} n \ neq -1 {\ mbox {)}}}

{\ displaystyle \ int \ cosh ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n + 1)}} \ sinh cx \ cosh ^ {n + 1} cx- {\ frac {n + 2} {n + 1}} \ int \ cosh ^ {n + 2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n <0 {\ mbox { ,}} n \ neq -1 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|\tanh {\frac {cx}{2}}\right|

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | \ tanh {\ frac {cx} {2}} \ dreapta |}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | \ tanh {\ frac {cx} {2}} \ dreapta |}

de asemenea:

\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\cosh cx-1}{\sinh cx}}\right|

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ cosh cx-1} {\ sinh cx}} \ right |}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ cosh cx-1} {\ sinh cx}} \ right |}

de asemenea:

\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\sinh cx}{\cosh cx+1}}\right|

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ sinh cx} {\ cosh cx +1}} \ dreapta |}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ sinh cx} {\ cosh cx +1}} \ dreapta |}

de asemenea:

\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\log \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ cosh cx-1} {\ cosh cx + 1}} \ right |}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh cx}} = {\ frac {1} {c}} \ log \ left | {\ frac {\ cosh cx-1} {\ cosh cx + 1}} \ right |}

\int {\frac {\mathrm {d} x}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ cosh cx}} = {\ frac {2} {c}} \ arctan e ^ {cx}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ cosh cx}} = {\ frac {2} {c}} \ arctan e ^ {cx}}

\int {\frac {\mathrm {d} x}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sinh ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh ^ {n} cx}} = {\ frac {\ cosh cx} {c (n-1) \ sinh ^ {n-1} cx}} - {\ frac {n-2} {n-1}} \ int {\ frac {\ mathrm {d} x} {\ sinh ^ {n-2} cx}} \ qquad {\ mbox {( pentru}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ sinh ^ {n} cx}} = {\ frac {\ cosh cx} {c (n-1) \ sinh ^ {n-1} cx}} - {\ frac {n-2} {n-1}} \ int {\ frac {\ mathrm {d} x} {\ sinh ^ {n-2} cx}} \ qquad {\ mbox {( pentru}} n \ neq 1 {\ mbox {)}}}

\int {\frac {\mathrm {d} x}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cosh ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ cosh ^ {n} cx}} = {\ frac {\ sinh cx} {c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {n-2} {n-1}} \ int {\ frac {\ mathrm {d} x} {\ cosh ^ {n-2} cx}} \ qquad {\ mbox {( pentru}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ mathrm {d} x} {\ cosh ^ {n} cx}} = {\ frac {\ sinh cx} {c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {n-2} {n-1}} \ int {\ frac {\ mathrm {d} x} {\ cosh ^ {n-2} cx}} \ qquad {\ mbox {( pentru}} n \ neq 1 {\ mbox {)}}}

\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq n{\mbox{)}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = {\ frac {\ cosh ^ {n-1} cx} {c (nm) \ sinh ^ {m-1} cx}} + {\ frac {n-1} {nm}} \ int {\ frac {\ cosh ^ {n-2} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq n {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = {\ frac {\ cosh ^ {n-1} cx} {c (nm) \ sinh ^ {m-1} cx}} + {\ frac {n-1} {nm}} \ int {\ frac {\ cosh ^ {n-2} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq n {\ mbox {)}}}

de asemenea:

\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = - {\ frac {\ cosh ^ {n + 1} cx} { c (m-1) \ sinh ^ {m-1} cx}} + {\ frac {nm + 2} {m-1}} \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = - {\ frac {\ cosh ^ {n + 1} cx} { c (m-1) \ sinh ^ {m-1} cx}} + {\ frac {nm + 2} {m-1}} \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq 1 {\ mbox {)}}}

de asemenea:

\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = - {\ frac {\ cosh ^ {n-1} cx} { c (m-1) \ sinh ^ {m-1} cx}} + {\ frac {n-1} {m-1}} \ int {\ frac {\ cosh ^ {n-2} cx} {\ sinh ^ {m-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ cosh ^ {n} cx} {\ sinh ^ {m} cx}} \ mathrm {d} x = - {\ frac {\ cosh ^ {n-1} cx} { c (m-1) \ sinh ^ {m-1} cx}} + {\ frac {n-1} {m-1}} \ int {\ frac {\ cosh ^ {n-2} cx} {\ sinh ^ {m-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq 1 {\ mbox {)}}}

\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}\mathrm {d} x\qquad {\mbox{(per }}m\neq n{\mbox{)}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = {\ frac {\ sinh ^ {m-1} cx} {c (mn) \ cosh ^ {n-1} cx}} + {\ frac {m-1} {mn}} \ int {\ frac {\ sinh ^ {m-2} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq n {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = {\ frac {\ sinh ^ {m-1} cx} {c (mn) \ cosh ^ {n-1} cx}} + {\ frac {m-1} {mn}} \ int {\ frac {\ sinh ^ {m-2} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} m \ neq n {\ mbox {)}}}

de asemenea:

\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = {\ frac {\ sinh ^ {m + 1} cx} {c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {mn + 2} {n-1}} \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ { n-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = {\ frac {\ sinh ^ {m + 1} cx} {c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {mn + 2} {n-1}} \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ { n-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

de asemenea:

\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = - {\ frac {\ sinh ^ {m-1} cx} { c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {m-1} {n-1}} \ int {\ frac {\ sinh ^ {m-2} cx} {\ cosh ^ {n-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int {\ frac {\ sinh ^ {m} cx} {\ cosh ^ {n} cx}} \ mathrm {d} x = - {\ frac {\ sinh ^ {m-1} cx} { c (n-1) \ cosh ^ {n-1} cx}} + {\ frac {m-1} {n-1}} \ int {\ frac {\ sinh ^ {m-2} cx} {\ cosh ^ {n-2} cx}} \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int x\sinh cx\,\mathrm {d} x={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx

{\ displaystyle \ int x \ sinh cx \, \ mathrm {d} x = {\ frac {1} {c}} x \ cosh cx - {\ frac {1} {c ^ {2}}} \ sinh cx }

{\ displaystyle \ int x \ sinh cx \, \ mathrm {d} x = {\ frac {1} {c}} x \ cosh cx - {\ frac {1} {c ^ {2}}} \ sinh cx }

\int x\cosh cx\,\mathrm {d} x={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx

{\ displaystyle \ int x \ cosh cx \, \ mathrm {d} x = {\ frac {1} {c}} x \ sinh cx - {\ frac {1} {c ^ {2}}} \ cosh cx }

{\ displaystyle \ int x \ cosh cx \, \ mathrm {d} x = {\ frac {1} {c}} x \ sinh cx - {\ frac {1} {c ^ {2}}} \ cosh cx }

\int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\log |\cosh cx|

{\ displaystyle \ int \ tanh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ log | \ cosh cx |}

{\ displaystyle \ int \ tanh cx \, \ mathrm {d} x = {\ frac {1} {c}} \ log | \ cosh cx |}

\int \coth cx\,\mathrm {d} x={\frac {1}{c}}\log |\sinh cx|

{\ displaystyle \ int \ coth cx \, \ mathrm {d} x = {\ frac {1} {c}} \ log | \ sinh cx |}

{\ displaystyle \ int \ coth cx \, \ mathrm {d} x = {\ frac {1} {c}} \ log | \ sinh cx |}

\int \tanh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int \ tanh ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n-1)}} \ tanh ^ {n-1} cx + \ int \ tanh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int \ tanh ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n-1)}} \ tanh ^ {n-1} cx + \ int \ tanh ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int \coth ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(per }}n\neq 1{\mbox{)}}

{\ displaystyle \ int \ coth ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n-1)}} \ coth ^ {n-1} cx + \ int \ coth ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

{\ displaystyle \ int \ coth ^ {n} cx \, \ mathrm {d} x = - {\ frac {1} {c (n-1)}} \ coth ^ {n-1} cx + \ int \ coth ^ {n-2} cx \, \ mathrm {d} x \ qquad {\ mbox {(for}} n \ neq 1 {\ mbox {)}}}

\int \sinh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}

{\ displaystyle \ int \ sinh bx \ sinh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh cx \ cosh bx-c \ cosh cx \ sinh bx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

{\ displaystyle \ int \ sinh bx \ sinh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh cx \ cosh bx-c \ cosh cx \ sinh bx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

\int \cosh bx\cosh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}

{\ displaystyle \ int \ cosh bx \ cosh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh bx \ cosh cx-c \ sinh cx \ cosh bx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

{\ displaystyle \ int \ cosh bx \ cosh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh bx \ cosh cx-c \ sinh cx \ cosh bx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

\int \cosh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}

{\ displaystyle \ int \ cosh bx \ sinh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh bx \ sinh cx-c \ cosh bx \ cosh cx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

{\ displaystyle \ int \ cosh bx \ sinh cx \, \ mathrm {d} x = {\ frac {1} {b ^ {2} -c ^ {2}}} (b \ sinh bx \ sinh cx-c \ cosh bx \ cosh cx) \ qquad {\ mbox {(per}} b ^ {2} \ neq c ^ {2} {\ mbox {)}}}

\int \sinh(ax+b)\sin(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)

{\ displaystyle \ int \ sinh (ax + b) \ sin (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ sin (cx + d) - {\ frac {c} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ cos (cx + d)}

{\ displaystyle \ int \ sinh (ax + b) \ sin (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ sin (cx + d) - {\ frac {c} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ cos (cx + d)}

\int \sinh(ax+b)\cos(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)

{\ displaystyle \ int \ sinh (ax + b) \ cos (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ cos (cx + d) + {\ frac {c} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ sin (cx + d)}

{\ displaystyle \ int \ sinh (ax + b) \ cos (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ cos (cx + d) + {\ frac {c} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ sin (cx + d)}

\int \cosh(ax+b)\sin(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)

{\ displaystyle \ int \ cosh (ax + b) \ sin (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ sin (cx + d) - {\ frac {c} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ cos (cx + d)}

{\ displaystyle \ int \ cosh (ax + b) \ sin (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ sin (cx + d) - {\ frac {c} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ cos (cx + d)}

\int \cosh(ax+b)\cos(cx+d)\,\mathrm {d} x={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)

{\ displaystyle \ int \ cosh (ax + b) \ cos (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ cos (cx + d) + {\ frac {c} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ sin (cx + d)}

{\ displaystyle \ int \ cosh (ax + b) \ cos (cx + d) \, \ mathrm {d} x = {\ frac {a} {a ^ {2} + c ^ {2}}} \ sinh (ax + b) \ cos (cx + d) + {\ frac {c} {a ^ {2} + c ^ {2}}} \ cosh (ax + b) \ sin (cx + d)}

Bibliografie

Murray R. Spiegel, Manual de matematică , Etas Libri, 1974, pp. 86-92.

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Tabelele integrale