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Integration of a trig function > 0

  1. Dec 7, 2012 #1
    I need to take the integral of the following function over one cycle.

    [itex]y(t) = sin(wt)cos(wt-p)[/itex]

    p is a phase angle that exist from 0 to pi. w is a frequency.

    This would be pretty straightforward, but I need to take the integral only of the portions of the function that are greater than zero.
    I need to do this for an efficiency analysis, where the negative portion of the cycle is waste energy and don't contribute to average power. I need an symbolic expression.

    The zero crossings are at t1 = pi/w and t2 = pi/(2w) + p/w which I had been using as limits of integration.

    [itex]\int ^{t_2}_{0} y(t) dt - \int ^{t_2}_{t_1} y(t) dt[/itex]

    The resulting expression is correct for values of for 0<=p <= pi/2. Above p = pi/2 though, the expression does not give me the result I need, because for p> pi/2, t2<t1 and the result of

    [itex] \int ^{t_2}_{t_1} y(t) dt[/itex]

    changes sign.

    I am not sure how to get around this. Any help would be...helpful. Thanks!
    Last edited: Dec 7, 2012
  2. jcsd
  3. Dec 7, 2012 #2


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    Science Advisor

    I would suggest an alternate approach. cos(wt-p) = coswtcosp + sinwtsinp, so you can then have two integrals from 0 to 2π/w. cosp∫sinwtcoswtdt + sinp∫sin2wtdt.
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