I need more help with this 1/(g-(k/m)v^2) dv?

I need to make a paper explaining how to solve this integral I need to factor fractions and end with a hyperbolic trig function. I will apreciate any help thanks.

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  • Dr D
    Lv 7
    1 decade ago
    Favorite Answer

    Essentially you want to write the integral as:

    (m/k)*dv / [ mg/k - v^2]

    And replace mg/k with a^2.

    The easiest method is to get the hyperbolic trig function directly from the integration.

    Use the substitution

    x = a*tanhθ

    dx = a*sech^2 θ

    a^2 - x^2 = a^2 * (1 - tanh^2 θ) = a^2 * sech^2 θ

    So the integral of dx / (a^2 - x^2) becomes

    integral of (1/a) dθ

    which gives you (1/a) θ = (1/a) * tanh^-1 (x/a)

    Now you just need to replace a with sqrt(mg/k) and x with v.

    ------------

    If you wish to integrate using partial fractions, then convert to hyperbolic trig, check out the links below where I answered very similar questions in the past.

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