Clever Geek Handbook
📜 ⬆️ ⬇️

List of integrals of hyperbolic functions

List of integrals ( primitive functions) of hyperbolic functions . In the list, the additive integration constant is omitted everywhere.

∫sh⁡cxdx=onecch⁡cx{\ displaystyle \ int \ operatorname {sh} cx \, dx = {\ frac {1} {c}} \ operatorname {ch} cx} \ int \ operatorname {sh} cx \, dx = {\ frac {1} {c}} \ operatorname {ch} cx
∫ch⁡cxdx=onecsh⁡cx{\ displaystyle \ int \ operatorname {ch} cx \, dx = {\ frac {1} {c}} \ operatorname {sh} cx} \ int \ operatorname {ch} cx \, dx = {\ frac {1} {c}} \ operatorname {sh} cx
∫sh2⁡cxdx=onefourcsh⁡2cx-x2{\ displaystyle \ int \ operatorname {sh} ^ {2} cx \, dx = {\ frac {1} {4c}} \ operatorname {sh} 2cx - {\ frac {x} {2}}} \ int \ operatorname {sh} ^ {2} cx \, dx = {\ frac {1} {4c}} \ operatorname {sh} 2cx - {\ frac {x} {2}}
∫ch2⁡cxdx=onefourcsh⁡2cx+x2{\ displaystyle \ int \ operatorname {ch} ^ {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}}
∫shn⁡cxdx=onecnshn-one⁡cxch⁡cx-n-onen∫shn-2⁡cxdx(n>0){\ 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 \ qquad {\ mbox {(}} n> 0 {\ mbox {)}}} \ 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 {)}}
also:∫shn⁡cxdx=onec(n+one)shn+one⁡cxch⁡cx-n+2n+one∫shn+2⁡cxdx(n<0,n≠-one) {\ 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 \ qquad {\ mbox {(}} n <0 {\ mbox { ,}} n \ neq -1 {\ mbox {)}}} \ 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 {)}}
∫chn⁡cxdx=onecnsh⁡cxchn-one⁡cx+n-onen∫chn-2⁡cxdx(n>0){\ 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 \ qquad {\ mbox {(}} n> 0 {\ 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 {)}}
also:∫chn⁡cxdx=-onec(n+one)sh⁡cxchn+one⁡cx-n+2n+one∫chn+2⁡cxdx(n<0,n≠-one) {\ 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 \ qquad {\ mbox {(}} n <0 {\ mbox {,}} n \ neq -1 {\ mbox {)}}} \ 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 {)}}
∫dxsh⁡cx=onecln⁡|th⁡cx2|=onecln⁡|ch⁡cx-onesh⁡cx|=onecln⁡|sh⁡cxch⁡cx+one|=onecln⁡|ch⁡cx-onech⁡cx+one|{\ displaystyle \ 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} 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 |
∫dxsh2⁡cx=-oneccth⁡cx{\ displaystyle \ int {\ frac {dx} {\ operatorname {sh} ^ {2} cx}} = - {\ frac {1} {c}} \ operatorname {cth} cx} \ int {\ frac {dx} {\ operatorname {sh} ^ {2} cx}} = - {\ frac {1} {c}} \ operatorname {cth} cx
∫dxch⁡cx=2carctg⁡ecx{\ displaystyle \ int {\ frac {dx} {\ operatorname {ch} cx}} = {\ frac {2} {c}} \ operatorname {arctg} e ^ {cx}} \ int {\ frac {dx} {\ operatorname {ch} cx}} = {\ frac {2} {c}} \ operatorname {arctg} e ^ {{cx}}
∫dxch2⁡cx=onecth⁡cx{\ displaystyle \ int {\ frac {dx} {\ operatorname {ch} ^ {2} cx}} = {\ frac {1} {c}} \ operatorname {th} cx} \ int {\ frac {dx} {\ operatorname {ch} ^ {2} cx}} = {\ frac {1} {c}} \ operatorname {th} cx
∫dxshn⁡cx=ch⁡cxc(n-one)shn-one⁡cx-n-2n-one∫dxshn-2⁡cx(n≠one){\ 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 \ neq 1 {\ mbox {)}}} \ 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 {)}}
∫dxchn⁡cx=sh⁡cxc(n-one)chn-one⁡cx+n-2n-one∫dxchn-2⁡cx(n≠one){\ 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 \ 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 {)}}
∫chn⁡cxshm⁡cxdx=chn-one⁡cxc(n-m)shm-one⁡cx+n-onen-m∫chn-2⁡cxshm⁡cxdx(m≠n){\ displaystyle \ int {\ frac {\ operatorname {ch} ^ {n} cx} {\ operatorname {sh} ^ {m} cx}} dx = {\ frac {\ operatorname {ch} ^ {n-1} cx} {c (nm) \ operatorname {sh} ^ {m-1} cx}} + {\ frac {n-1} {nm}} \ int {\ frac {\ operatorname {ch} ^ {n-2 } cx} {\ operatorname {sh} ^ {m} cx}} dx \ qquad {\ mbox {(}} m \ neq n {\ mbox {)}}} \ int {\ frac {\ operatorname {ch} ^ {n} cx} {\ operatorname {sh} ^ {m} cx}} dx = {\ frac {\ operatorname {ch} ^ {{n-1}} cx } {c (nm) \ operatorname {sh} ^ {{m-1}} cx}} + {\ frac {n-1} {nm}} \ int {\ frac {\ operatorname {ch} ^ {{n -2}} cx} {\ operatorname {sh} ^ {m} cx}} dx \ qquad {\ mbox {(}} m \ neq n {\ mbox {)}}
also:∫chn⁡cxshm⁡cxdx=-chn+one⁡cxc(m-one)shm-one⁡cx+n-m+2m-one∫chn⁡cxshm-2⁡cxdx(m≠one) {\ 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 \ qquad {\ mbox {(}} m \ neq 1 {\ mbox {)}}} \ 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 {)}}
also:∫chn⁡cxshm⁡cxdx=-chn-one⁡cxc(m-one)shm-one⁡cx+n-onem-one∫chn-2⁡cxshm-2⁡cxdx(m≠one) {\ 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 \ qquad {\ mbox {(}} m \ neq 1 {\ mbox {)}}} \ 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 {)}}
∫shm⁡cxchn⁡cxdx=shm-one⁡cxc(m-n)chn-one⁡cx+m-onem-n∫shm-2⁡cxchn⁡cxdx(m≠n){\ displaystyle \ int {\ frac {\ operatorname {sh} ^ {m} cx} {\ operatorname {ch} ^ {n} cx}} dx = {\ frac {\ operatorname {sh} ^ {m-1} cx} {c (mn) \ operatorname {ch} ^ {n-1} cx}} + {\ frac {m-1} {mn}} \ int {\ frac {\ operatorname {sh} ^ {m-2 } cx} {\ operatorname {ch} ^ {n} cx}} dx \ qquad {\ mbox {(}} m \ neq n {\ mbox {)}}} \ int {\ frac {\ operatorname {sh} ^ {m} cx} {\ operatorname {ch} ^ {n} cx}} dx = {\ frac {\ operatorname {sh} ^ {{m-1}} cx } {c (mn) \ operatorname {ch} ^ {{n-1}} cx}} + {\ frac {m-1} {mn}} \ int {\ frac {\ operatorname {sh} ^ {{m -2}} cx} {\ operatorname {ch} ^ {n} cx}} dx \ qquad {\ mbox {(}} m \ neq n {\ mbox {)}}
also:∫shm⁡cxchn⁡cxdx=shm+one⁡cxc(n-one)chn-one⁡cx+m-n+2n-one∫shm⁡cxchn-2⁡cxdx(n≠one) {\ 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 \ qquad {\ mbox {(}} n \ neq 1 {\ mbox {)}}} \ 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 {)}}
also:∫shm⁡cxchn⁡cxdx=-shm-one⁡cxc(n-one)chn-one⁡cx+m-onen-one∫shm-2⁡cxchn-2⁡cxdx(n≠one) {\ 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 \ qquad {\ mbox {(}} n \ neq 1 {\ mbox {)}}} \ 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 {)}}
∫xsh⁡cxdx=onecxch⁡cx-onec2sh⁡cx{\ displaystyle \ int x \ operatorname {sh} cx \, dx = {\ frac {1} {c}} x \ operatorname {ch} cx - {\ frac {1} {c ^ {2}}} \ operatorname {sh} cx} \ int x \ operatorname {sh} cx \, dx = {\ frac {1} {c}} x \ operatorname {ch} cx - {\ frac {1} {c ^ {2}}} \ operatorname {sh} cx
∫xch⁡cxdx=onecxsh⁡cx-onec2ch⁡cx{\ displaystyle \ int x \ operatorname {ch} cx \, dx = {\ frac {1} {c}} x \ operatorname {sh} cx - {\ frac {1} {c ^ {2}}} \ operatorname {ch} cx} \ int x \ operatorname {ch} cx \, dx = {\ frac {1} {c}} x \ operatorname {sh} cx - {\ frac {1} {c ^ {2}}} \ operatorname {ch} cx
∫th⁡cxdx=onecln⁡|ch⁡cx|{\ displaystyle \ int \ operatorname {th} cx \, dx = {\ frac {1} {c}} \ ln | \ operatorname {ch} cx |} \ int \ operatorname {th} cx \, dx = {\ frac {1} {c}} \ ln | \ operatorname {ch} cx |
∫cth⁡cxdx=onecln⁡|sh⁡cx|{\ displaystyle \ int \ operatorname {cth} cx \, dx = {\ frac {1} {c}} \ ln | \ operatorname {sh} cx |} \ int \ operatorname {cth} cx \, dx = {\ frac {1} {c}} \ ln | \ operatorname {sh} cx |
∫th2⁡cxdx=x-onecth⁡cx{\ displaystyle \ int \ operatorname {th} ^ {2} cx \, dx = x - {\ frac {1} {c}} \ operatorname {th} cx} \ int \ operatorname {th} ^ {2} cx \, dx = x - {\ frac {1} {c}} \ operatorname {th} cx
∫cth2⁡cxdx=x-oneccth⁡cx{\ displaystyle \ int \ operatorname {cth} ^ {2} cx \, dx = x - {\ frac {1} {c}} \ operatorname {cth} cx} \ int \ operatorname {cth} ^ {2} cx \, dx = x - {\ frac {1} {c}} \ operatorname {cth} cx
∫thn⁡cxdx=-onec(n-one)thn-one⁡cx+∫thn-2⁡cxdx(n≠one){\ displaystyle \ 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 {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 {)}}
∫cthn⁡cxdx=-onec(n-one)cthn-one⁡cx+∫cthn-2⁡cxdx(n≠one){\ displaystyle \ 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 {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 {)}}
∫sh⁡bxsh⁡cxdx=oneb2-c2(bsh⁡cxch⁡bx-cch⁡cxsh⁡bx)(b2≠c2){\ 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) \ qquad {\ mbox {(}} b ^ {2} \ neq c ^ {2} {\ 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 {)}}
∫ch⁡bxch⁡cxdx=oneb2-c2(bsh⁡bxch⁡cx-csh⁡cxch⁡bx)(b2≠c2){\ 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) \ 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 {)}}
∫ch⁡bxsh⁡cxdx=oneb2-c2(bsh⁡bxsh⁡cx-cch⁡bxch⁡cx)(b2≠c2){\ 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) \ 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 {)}}
∫sh⁡(ax+b)sin⁡(cx+d)dx=aa2+c2ch⁡(ax+b)sin⁡(cx+d)-ca2+c2sh⁡(ax+b)cos⁡(cx+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 )} \ 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⁡(ax+b)cos⁡(cx+d)dx=aa2+c2ch⁡(ax+b)cos⁡(cx+d)+ca2+c2sh⁡(ax+b)sin⁡(cx+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 )} \ 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⁡(ax+b)sin⁡(cx+d)dx=aa2+c2sh⁡(ax+b)sin⁡(cx+d)-ca2+c2ch⁡(ax+b)cos⁡(cx+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 )} \ 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⁡(ax+b)cos⁡(cx+d)dx=aa2+c2sh⁡(ax+b)cos⁡(cx+d)+ca2+c2ch⁡(ax+b)sin⁡(cx+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 )} \ 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)


Bibliography

Books
  • Gradshtein I. S. Ryzhik I. M. Tables of integrals, sums, series and products. - 4th ed. - M .: Nauka, 1963. - ISBN 0-12-294757-6 // EqWorld
  • Dvayt G. B. Tables of Integrals of St. Petersburg: Publishing House and Printing House of VNIIG im. B.V. Vedeneeva, 1995 .-- 176 p. - ISBN 5-85529-029-8 .
  • D. Zwillinger. CRC Standard Mathematical Tables and Formulae , 31st ed., 2002. ISBN 1-58488-291-3 .
  • M. Abramowitz and IA Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964. ISBN 0-486-61272-4
  • Korn G.A., Korn T.M. Math reference book for scientists and engineers . - M .: " Science ", 1974.
Integral Tables
  • Integrals at EqWorld
  • SOS Mathematics: Tables and Formulas
Calculation of Integrals
  • The Integrator (at Wolfram Research )
  • Empire of Numbers


Source - https://ru.wikipedia.org/w/index.php?title=List of_integrals_of_ hyperbolic_functions&oldid = 99431228


More articles:

  • Half Plane
  • Archdiocese
  • Syngonia
  • Janissaries
  • Muhali
  • Slovak Extra League
  • Zulus
  • Sutra
  • O'Sullivan Ronnie
  • Stepan

All articles

Clever Geek | 2019