Calculus 2

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integral of sinh(p*ln(x/xsub0))dx

  • Calculus 2 -

    using a rather than x0 for ease of typing,

    ∫sinh(p ln(x/a)) dx

    recall that sinh(u) = (e^u - e^-u)/2
    so, since
    e^(p ln(x/a)) = (e^(ln(x/a)))^p = (x/a)^p
    and, since e^-u = 1/e^u,
    e^(-p ln(x/a)) = (a/x)^p

    sinh(p ln(x/a)) = 1/2 (x^p/a^p - a^p/x^p)

    integrate that to get

    1/2 (x^(p+1)/((p+1)(a^p)) + a^p/(p-1)x^(p-1)))

    (p-1)x^(2p+1) + (p+1)xa^(2p)
    --------------------------------------
    2(p^2-1) a^p x^p

    you can play around with that if you want. If you get really ambitious, you might even end up with

    letting u = p*ln(x/a),

    x(p cosh(u) - sinh(u))/(p^2-1)

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