DLMF:4.40.E6 (Q1978)

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DLMF:4.40.E6
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    Statements

    Defining formula
    ∫ coth ⁡ x ⁢ d x = ln ⁡ ( sinh ⁡ x ) , hyperbolic-cotangent 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\int\coth x\mathrm{d}x=\ln\left(\sinh x\right),}}
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    DLMF id
    DLMF:4.40.E6
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    constraint
    0 < x < ∞ 0 𝑥 {\displaystyle{\displaystyle 0<x<\infty}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
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