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Math Help - Hyperbolic functions

  1. #1
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    Hyperbolic functions

    Evaluate:

    \int\tanh 2x\, ln(cosh 2x)\, dx

    \int\ x^2, csch^2\, x^3\, dx

    \int\ x cosh x^2\, sinh x^2\, dx
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  2. #2
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    \int \tanh(2x)\ln(\cosh(2x))dx

    let u = \ln(\cosh(2x))

    du = \frac{1}{\sinh(2x)}.2\cosh(2x) <<< using chain rule derivative

    substitute du and cancellation
    \int \frac{\sinh{2x}}{\cosh{2x}}u\frac{\cosh{2x}}{2\sin  h{2x}}du


    take 1/2 out
    \frac{1}{2}\int u du

    integrate
    \frac{1}{2} (\frac{1}{2}u^2)
    substitute u = \ln(\cosh(2x))

    \frac{1}{4} \ln^2(\cosh{2x}) + C
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  3. #3
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    \int x^2 csch^2{x^3} dx

    let u = x^3
    du = 3x^2

    \int csch^2{u} \frac{du}{3}

    integrate and
    substitute u
    -\frac{1}{3}\coth{x^3} + C
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  4. #4
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    \int x \cosh x^2, \sinh x^2 dx

    let u = x^2
    du = 2xdx
    \int \sinh{u} \cosh{u} \frac{du}{2}

    let y = sinh{u}
    dy = cosh{u}

    \frac{1}{2}\int y dy

    \frac{1}{2} (\frac{y^2}{2})

    from y = sinh{u}

    \frac{1}{4} \sinh^2{u} + C

    \frac{1}{4} \sinh^2{x^2} + C
    Last edited by ^_^Engineer_Adam^_^; December 7th 2007 at 07:13 AM.
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