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Math Help - Definite integral of hyperbolic function

  1. #1
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    Definite integral of hyperbolic function

    If \displaystyle I_n=\int^1_0\frac{\sinh^{2n}x}{\cosh x} dx, prove that

    \displaystyle I_{n+1}+I_n=\frac{\sinh^{2n+1} (1)}{2n+1}

    Show that I_0=-\frac{\pi}{2}+2\arctan e and evaluate I_3.

    My problem is showing that I_0=-\frac{\pi}{2}+2\arctan e.
    \displaystyle I_0=\int^1_0\frac{\sinh^0 x}{\cosh x}dx
    \displaystyle =\int^1_0\frac{1}{\cosh x} dx
    \displaystyle =\left[\frac{1}{\sinh x} \ln|\cosh x|\right]^1_0
    Problem is after substituting 1 and 0, i still can't get -\frac{\pi}{2}+2\arctan e
    Thanks!
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  2. #2
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    cosh(x) = \frac{e^x + e^{-x}}{2}

    I_o = \int^1_0\frac{2}{e^x+e^{-}x}dx

    I_o = \int^1_0\frac{2e^x}{e^{2x} + 1}dx

    Substitute e^x = t and solve the integration.
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  3. #3
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    cosh(x) = \frac{e^x + e^{-x}}{2}

    I_o = \int^1_0\frac{2}{e^x+e^{-x}}dx

    I_o = \int^1_0\frac{2e^x}{e^{2x} + 1}dx

    Substitute e^x = t and solve the integration.
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