Lista de integrales de funciones hiperbólicas inversas

La siguiente es una lista de integrales indefinidas (antiderivadas) de expresiones que involucran funciones hiperbólicas inversas. Para obtener una lista completa de fórmulas integrales, consulte las listas de integrales.

• En todas las fórmulas la constante a se supone que no es cero, y C denota la constante integración.
• Para cada fórmula de integración hiperbólica inversa hay una fórmula correspondiente en la lista de integrales de funciones trigonométricas inversas.
• El estándar ISO 80000-2 utiliza el prefijo "ar-" en lugar de "arc-" para las funciones hiperbólicas inversas; lo hacemos aquí.

Fórmulas de integración del seno hiperbólico inverso

∫ ∫ arsinh⁡ ⁡ ()ax)dx=xarsinh⁡ ⁡ ()ax)− − a2x2+1a+C{displaystyle int operatorname {arsinh} (ax),dx=xoperatorname {arsinh} (ax)-{frac {sqrt {a^{2}x^{2}+1}{a}}}+C}}

∫ ∫ xarsinh⁡ ⁡ ()ax)dx=x2arsinh⁡ ⁡ ()ax)2+arsinh⁡ ⁡ ()ax)4a2− − xa2x2+14a+C{displaystyle int xoperatorname {arsinh},dx={frac {x^{2}operatorname {arsinh} {2}}{2}}+{frac {fnMicrosoft {f}{4a^{2}}}}-{frac {x{sqrt {2}x} {2}+1} {4a}+C}

∫ ∫ x2arsinh⁡ ⁡ ()ax)dx=x3arsinh⁡ ⁡ ()ax)3− − ()a2x2− − 2)a2x2+19a3+C{displaystyle int x^{2}operatorname {arsinh} (ax),dx={frac [x^{3}operatorname {arsinh} {3}}{3}-{frac {left(a^{2}x^{2}-2right){sqrt {cH00}}}+C}

∫ ∫ xmarsinh⁡ ⁡ ()ax)dx=xm+1arsinh⁡ ⁡ ()ax)m+1− − am+1∫ ∫ xm+1a2x2+1dx()mل ل − − 1){displaystyle int x^{m}operatorname {arsinh} (ax),dx={frac [x^{m+1}operatorname {arsinh} {m+1}-{frac {fn} {fn} {fnK} {fn}} {fn}} {fn}} {fn}}} {fn}}} {fn}}} {fn}}}}} {fn}} {fnfn}}}}nnfnf} {fn}}}}}nnnnnnnn}nnn}nnnn}}nn}n}n}n}n}n}n}nn}n}n}nnn}n}nnnnnnnnnnnnnnnnnnn}n}n}n}n}n}n}n}n}nnnnnn}n {a^{2}x^{2}+1},dxquad (mneq -1)}

∫ ∫ arsinh⁡ ⁡ ()ax)2dx=2x+xarsinh⁡ ⁡ ()ax)2− − 2a2x2+1arsinh⁡ ⁡ ()ax)a+C{displaystyle int operatorname {arsinh} (ax)^{2},dx=2x+xoperatorname {arsinh} (ax)^{2}-{frac {2{sqrt {a^{2}x^{2}+1}}operatorname {arsinh} {ax}{a}}}}+C} {}} {}}}} {}}}}}}}}}}}}}}}

∫ ∫ arsinh⁡ ⁡ ()ax)ndx=xarsinh⁡ ⁡ ()ax)n− − na2x2+1arsinh⁡ ⁡ ()ax)n− − 1a+n()n− − 1)∫ ∫ arsinh⁡ ⁡ ()ax)n− − 2dx{fnMicrosoft Sans Serif} {fnMicrosoft Sans Serif} {fnMicrosoft Sans Serif} {fn} {fn} {fnfn} {fn0}fn1}fnfnfn1}fnfncH009cH009cH009cH009}cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009cH009}cH009}cH009cH009cH009cH009cH009}cH009cH009cH009cH009cH009cH009cH009cH009

∫ ∫ arsinh⁡ ⁡ ()ax)ndx=− − xarsinh⁡ ⁡ ()ax)n+2()n+1)()n+2)+a2x2+1arsinh⁡ ⁡ ()ax)n+1a()n+1)+1()n+1)()n+2)∫ ∫ arsinh⁡ ⁡ ()ax)n+2dx()nل ل − − 1,− − 2){fnMicrosoft Sans Serif} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0} {fn0}cH00}cH00}}cH00}cH00}cH00}cH00cH00}cH00}cH00}cH00}cH00}cH00}cH00}cH00cH00}cH00cH00cH00}cH00}cH00cH00}cH00}cH00}cH00}cH00cH00cH00cH00}cH00}cH00}cH00cH00}cH00cH00cH00}cH00}c

Fórmulas de integración del coseno hiperbólico inverso

∫ ∫ arcosh⁡ ⁡ ()ax)dx=xarcosh⁡ ⁡ ()ax)− − ax+1ax− − 1a+C{displaystyle int operatorname {arcosh} (ax),dx=xoperatorname {arcosh} (ax)-{frac {sqrt {ax+1}{sqrt {fnK}} {fn}}}}} {fn}}}} {c}}}} {c}}}} {c}}}}}}}}} {c}}}}}}}}} {c}}}}}} {c}}}}}}}}} {}}}}}}}}}}}}}} {}}}}}}}}}}} {}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}} {}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

∫ ∫ xarcosh⁡ ⁡ ()ax)dx=x2arcosh⁡ ⁡ ()ax)2− − arcosh⁡ ⁡ ()ax)4a2− − xax+1ax− − 14a+C{displaystyle int xoperatorname {arcosh}(ax),dx={frac [x^{2}operatorname {arcosh} {2} {frac {fnMicroc {fnMicrosoft}{4a^{2}}}}}-{frac {x{sqrt {ax+1}{sqrt {ax-1}}}}} {4a}}}}}}} {c}}} {c}}} {c}}}}}}} {c}}}}}}}} {c}}}}}}}}}} {c}}}}}} {c}}}} {c}}}}}}}}}} {c}}}}}}}}}}}}} {c}}}}}} {c}}}}}}}}}}} {c} {c}}}}}}}}}}}}}}}}}} {c}}}}}} {c}}}}}}}}}}}}}}}}}

∫ ∫ x2arcosh⁡ ⁡ ()ax)dx=x3arcosh⁡ ⁡ ()ax)3− − ()a2x2+2)ax+1ax− − 19a3+C{displaystyle int x^{2}operatorname {arcosh} (ax),dx={frac [x^{3}operatorname {arcosh} {3}}{3}-{frac {left(a^{2}x^{2}+2right){sqrt {ax+1}{sqrt {ax-1}}{9a^{3}}}}}}}+C}

∫ ∫ xmarcosh⁡ ⁡ ()ax)dx=xm+1arcosh⁡ ⁡ ()ax)m+1− − am+1∫ ∫ xm+1ax+1ax− − 1dx()mل ل − − 1){displaystyle int x^{m}operatorname {arcosh} (ax),dx={frac [x^{m+1}operatorname {arcosh} {m+1}-{frac {a}{m+1} {fnK} {m+1}{sqrt {ax+1}{sqrt {ax-1}}}},dxquad (mneq -1)}} {fnK}}}}}} {\fnK}}}}}}} {fnKf}}}}}}}}}}}}}}}}}} {

∫ ∫ arcosh⁡ ⁡ ()ax)2dx=2x+xarcosh⁡ ⁡ ()ax)2− − 2ax+1ax− − 1arcosh⁡ ⁡ ()ax)a+C{fnMicrosoftware {fnMicrosoft Sans Serif} {fnMicrosoft Sans Serif} {fnMicrosoft Sans} {fnMicroc {2{sqrt {ax+1}{sqrt {ax-1}f}fnMicrosoft} {fnMicrosoft} {fnMicrosoft}}}}}}} {f}}}}}}}}}}}}}}}}}}}}}}}} {f} {f}f}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {

∫ ∫ arcosh⁡ ⁡ ()ax)ndx=xarcosh⁡ ⁡ ()ax)n− − nax+1ax− − 1arcosh⁡ ⁡ ()ax)n− − 1a+n()n− − 1)∫ ∫ arcosh⁡ ⁡ ()ax)n− − 2dx{displaystyle int operatorname {arcosh} (ax)^{n},dx=xoperatorname {arcosh} (ax)^{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}

∫ ∫ arcosh⁡ ⁡ ()ax)ndx=− − xarcosh⁡ ⁡ ()ax)n+2()n+1)()n+2)+ax+1ax− − 1arcosh⁡ ⁡ ()ax)n+1a()n+1)+1()n+1)()n+2)∫ ∫ arcosh⁡ ⁡ ()ax)n+2dx()nل ل − − 1,− − 2){displaystyle int operatorname {arcosh} (ax)^{n},dx=-{frac {xoperatorname {arcosh} (ax)^{n+2}{(n+1)(n+2)}+{fracfrac {sqrt {ax+1}{sqrt {ax-1}operatorname {arcosh} (ax)^{n+1}}{a(n+1)}}+{frac {1}{ n+1)}int operatorname {arcosh} (ax)}{n+2},dxquad (nne)

Fórmulas de integración de la tangente hiperbólica inversa

∫ ∫ Artanh⁡ ⁡ ()ax)dx=xArtanh⁡ ⁡ ()ax)+In⁡ ⁡ ()1− − a2x2)2a+C{displaystyle int operatorname {artanh} (ax),dx=xoperatorname {artanh} (ax)+{frac {ln left(1-a^{2}x^{2}right)}{2a}}+C}

∫ ∫ xArtanh⁡ ⁡ ()ax)dx=x2Artanh⁡ ⁡ ()ax)2− − Artanh⁡ ⁡ ()ax)2a2+x2a+C{displaystyle int xoperatorname {artanh}(ax),dx={frac [x^{2}operatorname {artanh} {2} {frac {fnMicroc {fnMicrosoft}{2a^{2}}}}}+{frac {x}{2a}+C}

∫ ∫ x2Artanh⁡ ⁡ ()ax)dx=x3Artanh⁡ ⁡ ()ax)3+In⁡ ⁡ ()1− − a2x2)6a3+x26a+C{displaystyle int x^{2}operatorname {artanh} (ax),dx={frac {x^{3}fnfnnfnncncncncncncncncncncnccnccH00}cH009}\ccccH0}cccH00cH00cH00}cH00cH00cH00cH00}}}\\\cH00cH00\\cH00cH00cH00cH00cH00cH00cH00\cH00cH00cH00cH00cH00cH00cH00cH00}cH00}cH00cH00cH00cH00cH00cH00}}}cH00\cH00ccH00cH00cH00}}}}ccH {x^{2}{6a}}+C}

∫ ∫ xmArtanh⁡ ⁡ ()ax)dx=xm+1Artanh⁡ ⁡ ()ax)m+1− − am+1∫ ∫ xm+11− − a2x2dx()mل ل − − 1){displaystyle int x^{m}operatorname {artanh} (ax),dx={frac [x^{m+1}operatorname {artanh} {m+1}-{frac {a}{m+1}int {fnMic} {x^{m+1}{1-a^{2}},dxquad (mneq -1)}

Fórmulas de integración de la cotangente hiperbólica inversa

∫ ∫ arco⁡ ⁡ ()ax)dx=xarco⁡ ⁡ ()ax)+In⁡ ⁡ ()a2x2− − 1)2a+C{displaystyle int operatorname {arcoth} (ax),dx=xoperatorname {arcoth} (ax)+{frac {ln left(a^{2}x^{2}-1right)}{2a}+C}

∫ ∫ xarco⁡ ⁡ ()ax)dx=x2arco⁡ ⁡ ()ax)2− − arco⁡ ⁡ ()ax)2a2+x2a+C{displaystyle int xoperatorname {arcoth}(ax),dx={frac [x^{2}operatorname {arcoth} (ax)}{2}-{frac {operatorname {arcoth} (ax)}{2a^{2}}}}+{frac {x}{2a}+C}

∫ ∫ x2arco⁡ ⁡ ()ax)dx=x3arco⁡ ⁡ ()ax)3+In⁡ ⁡ ()a2x2− − 1)6a3+x26a+C{displaystyle int x^{2}operatorname {arcoth} (ax),dx={frac {x^{3}fnfnfnfnncncncncncncncncncnccH00}cH009}\fn1fn1}mcH0}cccH0cH0cH0} {x^{2}{6a}}+C}

∫ ∫ xmarco⁡ ⁡ ()ax)dx=xm+1arco⁡ ⁡ ()ax)m+1+am+1∫ ∫ xm+1a2x2− − 1dx()mل ل − − 1){displaystyle int x^{m}operatorname {arcoth} (ax),dx={frac [x^{m+1}operatorname {arcoth} {m+1}+{frac {a}{m+1}int {fnMic} {x^{m+1}{2}x^{2},dxquad (mneq -1)}

Fórmulas de integración de la secante hiperbólica inversa

∫ ∫ arsech⁡ ⁡ ()ax)dx=xarsech⁡ ⁡ ()ax)− − 2aarctan⁡ ⁡ 1− − ax1+ax+C{displaystyle int operatorname {arsech} (ax),dx=xoperatorname {arsech} (ax)-{frac {2}{a}}operatorname {arctan} {sqrt {frac} {1-ax}{1+ax}}+C}

∫ ∫ xarsech⁡ ⁡ ()ax)dx=x2arsech⁡ ⁡ ()ax)2− − ()1+ax)2a21− − ax1+ax+C{displaystyle int xoperatorname {arsech}(ax),dx={frac {x^{2}fnMicroc {fnMicrosoft Sans Serif}{2}} {fnMicroc {fn} {fnMicroc} {fnMicroc} {fnMicroc} {fnMicroc} {fnMicroc} {fnMicroc}} {fnMicroc}} {f}}}}}} {f}}f}}}}}}f}f}f}f}f}f}fnKf}fnKfnKfnKfnKfnKfnKf}fnKfnKfnKfnKf}fnun}fnun}}fnun}fnKfnun}fnun}fnun}fnun}fnun}fnun}fnunfnun}fnun} {1-ax}{1+ax}}+C}

∫ ∫ x2arsech⁡ ⁡ ()ax)dx=x3arsech⁡ ⁡ ()ax)3− − 13a3arctan⁡ ⁡ 1− − ax1+ax− − x()1+ax)6a21− − ax1+ax+C{displaystyle int x^{2}operatorname {arsech} (ax),dx={frac {x^{3}fnMicroc {3} {3a}}fnMicroc {3a}}fnuncio {arctan} {sqrt {frac} {fnMicroc} {fnMicroc} {fn}fnMicroc} {fnMicroc} {fnMicroc} {f}f}}}}}f}}fnMicroc}}f}f}f}f}f}f}f}f}f}fnKfnKfnKfnKfnKfnKf}fnKfnKfnKfnKf}fnK}fnKf}fnKf}fnKf}fnKfnKfnKfnKfnKfnKf}fnK}f}fnK {1-ax}{1+ax}} {frac {x(1+ax)}{6a^{2}}{sqrt {frac} {1-ax}{1+ax}}+C}

∫ ∫ xmarsech⁡ ⁡ ()ax)dx=xm+1arsech⁡ ⁡ ()ax)m+1+1m+1∫ ∫ xm()1+ax)1− − ax1+axdx()mل ل − − 1){displaystyle int x^{m}operatorname {arsech} (ax),dx={frac {x^{m+1}fnMicroc {m+1}} {m+1}}int {fnMicroc {x^{m}{(1+ax){sqrt {frac {1-ax}{1+ax}}},dxquad (mneq -1)}

Fórmulas de integración de la cosecante hiperbólica inversa

∫ ∫ arcsch⁡ ⁡ ()ax)dx=xarcsch⁡ ⁡ ()ax)+1aarco⁡ ⁡ 1a2x2+1+C{displaystyle int operatorname {arcsch} (ax),dx=xoperatorname {arcsch} (ax)+{frac {1}{a}}operatorname {arcoth} {sqrt {{fracrt {frac {1}{2}x^{2}}}+1}+C}

∫ ∫ xarcsch⁡ ⁡ ()ax)dx=x2arcsch⁡ ⁡ ()ax)2+x2a1a2x2+1+C{displaystyle int xoperatorname {arcsch}(ax),dx={frac [x^{2}operatorname {arcsch}{2}}{2}{frac {x}{sqrt {frac}{sqrt {frac} {fnMic} {fnK} {fnK} {f}}} {fnK}} {fnKfnMic} {f}f}}}}}f}}}}}f}f}}f}f}f}f}f}f}f}fnKf}fnKf}fnKfnKfnKf}fnKf}f}fnKfnKfnKfnKfnKfnKfnKfnKfnKf}}fnKfnKfnKfnKf}}fnKf}fnKf}}}fn {1}{2}x^{2}}}+1}+C}

∫ ∫ x2arcsch⁡ ⁡ ()ax)dx=x3arcsch⁡ ⁡ ()ax)3− − 16a3arco⁡ ⁡ 1a2x2+1+x26a1a2x2+1+C{displaystyle int x^{2}operatorname {arcsch} (ax),dx={frac [x^{3}operatorname {arcsch} {3}} {frac {1}{6a^{3}}}operatorname {arcoth} {sqrt {frac {frac} {fnMicroc}} {f}}fnMicroc}} {fnMicroc} {1}{2}x^{2}}}+1}+{frac} {fnMicroc} {fnK}} {fnMicroc} {fnK} {f}} {fn}} {fnK}} {f}}} {fnK}} {fnf} {f}} {fnK}} {f}}} {fnf}}}}} {f} {f}f}}}}}}}}}}}}}}}}}}}} {f} {f} {f} {f}} {f}f}}f}}}}}f}f}}}}}}}}}}}}}} {f} {sqf}} {f}}} {sqf}} {f}f}f}f} {f} {f} {f}f}f}}}}f}f}}f}f}}}}}}} {1}{2}x^{2}}}+1}+C}

∫ ∫ xmarcsch⁡ ⁡ ()ax)dx=xm+1arcsch⁡ ⁡ ()ax)m+1+1a()m+1)∫ ∫ xm− − 11a2x2+1dx()mل ل − − 1){displaystyle int x^{m}operatorname {arcsch} (ax),dx={frac [x^{m+1}operatorname {arcsch} {m+1}}+{frac {1}{a(m+1)}}}int {frac {x^{m-1}{sqrt {frac}{sqrt {frac}}} {frac}}} {sq} {sq} {f}{f}} {f}f} {sqf}}} {f} {f} {f} {f} {f}f}f}f}f}f}f}f}f}f}f}f}f}f}f}f} {fnhf}fnf} {f}f} {fnhf}f}f} {f}fnKfnhfnhf}fnKf}fnhf}f}fn {1}{2}x^{2} 2}}+1}},dxquad (mneq -1)}