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Thread: Hyperbolic functions

  1. #1
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    Hyperbolic functions

    Evaluate:

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

    from y = sinh{u}

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

    $\displaystyle \frac{1}{4} \sinh^2{x^2} + C$
    Last edited by ^_^Engineer_Adam^_^; Dec 7th 2007 at 06:13 AM.
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