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# Danh sách tích phân với hàm hyperbolic ngược

Dưới đây là danh sách các tích phân với hàm hyperbolic ngược.

${\displaystyle \int \mathrm {arsinh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsinh} \,{\frac {x}{c}}-{\sqrt {x^{2}+c^{2}}}}$
${\displaystyle \int \mathrm {arcosh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcosh} \,{\frac {x}{c}}-{\sqrt {x^{2}-c^{2}}}}$
${\displaystyle \int \mathrm {artanh} \,{\frac {x}{c}}\,dx=x\,\mathrm {artanh} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |c^{2}-x^{2}|\qquad {\mbox{(}}|x|<|c|{\mbox{)}}}$
${\displaystyle \int \mathrm {arcoth} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcoth} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |x^{2}-c^{2}|\qquad {\mbox{(}}|x|>|c|{\mbox{)}}}$
${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {c^{2}-x^{2}}}}{c}}\qquad {\mbox{(}}x\in (0,\,c){\mbox{)}}}$
hay ${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}-2c\,\mathrm {arctan} \,{\sqrt {\frac {c-x}{c+x}}}}$
hay ${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}+2c\,\mathrm {arcsin} \,{\sqrt {\frac {x+c}{2c}}}}$
hay ${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}-c\,\mathrm {arctan} \,{\frac {x\,{\sqrt {\frac {c-x}{c+x}}}}{x-c}}}$
hay ${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}+c\,\mathrm {arcsin} \,{\frac {x}{c}}}$
hay ${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}-c\,\mathrm {arctan} \,{\sqrt {{\frac {c^{2}}{x^{2}}}-1}}}$
${\displaystyle \int \mathrm {arcsch} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {x^{2}+c^{2}}}}{c}}\qquad {\mbox{(}}x\in (0,\,c){\mbox{)}}}$
hay ${\displaystyle \int \mathrm {arcsch} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{c}}+c\,\mathrm {arcoth} \,{\sqrt {{\frac {c^{2}}{x^{2}}}+1}}}$
hay ${\displaystyle \int \mathrm {arcsch} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{c}}+c|\,\mathrm {arsinh} \,{\frac {x}{c}}|}$