Seznam integrálů inverzních hyperbolických funkcí - List of integrals of inverse hyperbolic functions

Následuje seznam neurčité integrály (antiderivativa ) výrazů zahrnujících inverzní hyperbolické funkce. Úplný seznam integrálních vzorců najdete v části seznamy integrálů.

• Ve všech vzorcích konstanta A se předpokládá, že je nenulová, a C označuje konstanta integrace.
• Pro každý vzorec inverzní hyperbolické integrace níže existuje odpovídající vzorec v seznam integrálů inverzních trigonometrických funkcí.

Inverzní hyperbolické sinusové integrační vzorce

${ displaystyle int operatorname {arsinh} (sekera) , dx = x operatorname {arsinh} (sekera) - { frac { sqrt {a ^ {2} x ^ {2} +1}} {a }} + C}$

${ displaystyle int x operatorname {arsinh} (sekera) , dx = { frac {x ^ {2} operatorname {arsinh} (ax)} {2}} + { frac { operatorname {arsinh} (sekera)} {4a ^ {2}}} - { frac {x { sqrt {a ^ {2} x ^ {2} +1}}} {4a}} + C}$

${ displaystyle int x ^ {2} operatorname {arsinh} (sekera) , dx = { frac {x ^ {3} operatorname {arsinh} (sekera)} {3}} - { frac { vlevo (a ^ {2} x ^ {2} -2 vpravo) { sqrt {a ^ {2} x ^ {2} +1}}} {9a ^ {3}}} + C}$

${ displaystyle int x ^ {m} operatorname {arsinh} (sekera) , dx = { frac {x ^ {m + 1} operatorname {arsinh} (sekera)} {m + 1}} - { frac {a} {m + 1}} int { frac {x ^ {m + 1}} { sqrt {a ^ {2} x ^ {2} +1}}} , dx quad ( m neq -1)}$

${ displaystyle int operatorname {arsinh} (sekera) ^ {2} , dx = 2x + x operatorname {arsinh} (sekera) ^ {2} - { frac {2 { sqrt {a ^ {2 } x ^ {2} +1}} operatorname {arsinh} (sekera)} {a}} + C}$

${ displaystyle int operatorname {arsinh} (sekera) ^ {n} , dx = x operatorname {arsinh} (sekera) ^ {n} - { frac {n { sqrt {a ^ {2} x ^ {2} +1}} operatorname {arsinh} (sekera) ^ {n-1}} {a}} + n (n-1) int operatorname {arsinh} (sekera) ^ {n-2} , dx}$

${ displaystyle int operatorname {arsinh} (sekera) ^ {n} , dx = - { frac {x operatorname {arsinh} (sekera) ^ {n + 2}} {(n + 1) (n +2)}} + { frac {{ sqrt {a ^ {2} x ^ {2} +1}} operatorname {arsinh} (ax) ^ {n + 1}} {a (n + 1) }} + { frac {1} {(n + 1) (n + 2)}} int operatorname {arsinh} (ax) ^ {n + 2} , dx quad (n neq -1, -2)}$

Inverzní hyperbolické kosinové integrační vzorce

${ displaystyle int operatorname {arcosh} (sekera) , dx = x operatorname {arcosh} (sekera) - { frac {{ sqrt {ax + 1}} { sqrt {ax-1}}} {a}} + C}$

${ displaystyle int x operatorname {arcosh} (sekera) , dx = { frac {x ^ {2} operatorname {arcosh} (ax)} {2}} - { frac { operatorname {arcosh} (ax)} {4a ^ {2}}} - { frac {x { sqrt {ax + 1}} { sqrt {ax-1}}} {4a}} + C}$

${ displaystyle int x ^ {2} operatorname {arcosh} (sekera) , dx = { frac {x ^ {3} operatorname {arcosh} (sekera)} {3}} - { frac { left (a ^ {2} x ^ {2} +2 right) { sqrt {ax + 1}} { sqrt {ax-1}}} {9a ^ {3}}} + C}$

${ displaystyle int x ^ {m} operatorname {arcosh} (sekera) , dx = { frac {x ^ {m + 1} operatorname {arcosh} (ax)} {m + 1}} - { frac {a} {m + 1}} int { frac {x ^ {m + 1}} {{ sqrt {ax + 1}} { sqrt {ax-1}}}} dx quad (m neq -1)}$

${ displaystyle int operatorname {arcosh} (sekera) ^ {2} , dx = 2x + x operatorname {arcosh} (sekera) ^ {2} - { frac {2 { sqrt {ax + 1} } { sqrt {ax-1}} operatorname {arcosh} (ax)} {a}} + C}$

${ displaystyle int operatorname {arcosh} (sekera) ^ {n} , dx = x operatorname {arcosh} (sekera) ^ {n} - { frac {n { sqrt {ax + 1}} { sqrt {ax-1}} operatorname {arcosh} (ax) ^ {n-1}} {a}} + n (n-1) int operatorname {arcosh} (ax) ^ {n-2} , dx}$

${ displaystyle int operatorname {arcosh} (sekera) ^ {n} , dx = - { frac {x operatorname {arcosh} (sekera) ^ {n + 2}} {(n + 1) (n +2)}} + { frac {{ sqrt {ax + 1}} { sqrt {ax-1}} operatorname {arcosh} (ax) ^ {n + 1}} {a (n + 1) }} + { frac {1} {(n + 1) (n + 2)}} int operatorname {arcosh} (ax) ^ {n + 2} , dx quad (n neq -1, -2)}$

Inverzní hyperbolické tangenciální integrační vzorce

${ displaystyle int operatorname {artanh} (sekera) , dx = x operatorname {artanh} (sekera) + { frac { ln left (1-a ^ {2} x ^ {2} right )} {2a}} + C}$

${ displaystyle int x operatorname {artanh} (sekera) , dx = { frac {x ^ {2} operatorname {artanh} (sekera)} {2}} - { frac { operatorname {artanh} (sekera)} {2a ^ {2}}} + { frac {x} {2a}} + C}$

${ displaystyle int x ^ {2} operatorname {artanh} (sekera) , dx = { frac {x ^ {3} operatorname {artanh} (sekera)} {3}} + { frac { ln left (1-a ^ {2} x ^ {2} right)} {6a ^ {3}}} + { frac {x ^ {2}} {6a}} + C}$

${ displaystyle int x ^ {m} operatorname {artanh} (sekera) , dx = { frac {x ^ {m + 1} operatorname {artanh} (sekera)} {m + 1}} - { frac {a} {m + 1}} int { frac {x ^ {m + 1}} {1-a ^ {2} x ^ {2}}} , dx quad (m neq - 1)}$

Inverzní hyperbolické kotangensové integrační vzorce

${ displaystyle int operatorname {arcoth} (sekera) , dx = x operatorname {arcoth} (sekera) + { frac { ln doleva (a ^ {2} x ^ {2} -1 doprava )} {2a}} + C}$

${ displaystyle int x operatorname {arcoth} (sekera) , dx = { frac {x ^ {2} operatorname {arcoth} (ax)} {2}} - { frac { operatorname {arcoth} (sekera)} {2a ^ {2}}} + { frac {x} {2a}} + C}$

${ displaystyle int x ^ {2} operatorname {arcoth} (sekera) , dx = { frac {x ^ {3} operatorname {arcoth} (sekera)} {3}} + { frac { ln left (a ^ {2} x ^ {2} -1 right)} {6a ^ {3}}} + { frac {x ^ {2}} {6a}} + C}$

${ displaystyle int x ^ {m} operatorname {arcoth} (sekera) , dx = { frac {x ^ {m + 1} operatorname {arcoth} (ax)} {m + 1}} + { frac {a} {m + 1}} int { frac {x ^ {m + 1}} {a ^ {2} x ^ {2} -1}} , dx quad (m neq - 1)}$

Inverzní hyperbolické sekanční integrační vzorce

${ displaystyle int operatorname {arsech} (sekera) , dx = x operatorname {arsech} (sekera) - { frac {2} {a}} operatorname {arctan} { sqrt { frac {1 -ax} {1 + ax}}} + C}$

${ displaystyle int x operatorname {arsech} (sekera) , dx = { frac {x ^ {2} operatorname {arsech} (sekera)} {2}} - { frac {(1 + sekera) } {2a ^ {2}}} { sqrt { frac {1-osa} {1 + ax}}} + C}$

${ displaystyle int x ^ {2} operatorname {arsech} (sekera) , dx = { frac {x ^ {3} operatorname {arsech} (sekera)} {3}} - { frac {1 } {3a ^ {3}}} operatorname {arctan} { sqrt { frac {1-ax} {1 + ax}}} - { frac {x (1 + ax)} {6a ^ {2} }} { sqrt { frac {1-osa} {1 + ax}}} + C}$

${ displaystyle int x ^ {m} operatorname {arsech} (sekera) , dx = { frac {x ^ {m + 1} operatorname {arsech} (ax)} {m + 1}} + { frac {1} {m + 1}} int { frac {x ^ {m}} {(1 + sekera) { sqrt { frac {1-sekera} {1 + sekera}}}}} , dx quad (m neq -1)}$

Inverzní hyperbolické vzorce integrace kosekans

${ displaystyle int operatorname {arcsch} (sekera) , dx = x operatorname {arcsch} (sekera) + { frac {1} {a}} operatorname {arcoth} { sqrt {{ frac { 1} {a ^ {2} x ^ {2}}} + 1}} + C}$

${ displaystyle int x operatorname {arcsch} (sekera) , dx = { frac {x ^ {2} operatorname {arcsch} (ax)} {2}} + { frac {x} {2a} } { sqrt {{ frac {1} {a ^ {2} x ^ {2}}} + 1}} + C}$

${ displaystyle int x ^ {2} operatorname {arcsch} (sekera) , dx = { frac {x ^ {3} operatorname {arcsch} (ax)} {3}} - { frac {1 } {6a ^ {3}}} operatorname {arcoth} { sqrt {{ frac {1} {a ^ {2} x ^ {2}}} + 1}} + { frac {x ^ {2 }} {6a}} { sqrt {{ frac {1} {a ^ {2} x ^ {2}}} + 1}} + C}$

${ displaystyle int x ^ {m} operatorname {arcsch} (sekera) , dx = { frac {x ^ {m + 1} operatorname {arcsch} (ax)} {m + 1}} + { frac {1} {a (m + 1)}} int { frac {x ^ {m-1}} { sqrt {{ frac {1} {a ^ {2} x ^ {2}} } +1}}} , dx quad (m neq -1)}$