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Math Help - Power Series Approximation

  1. #1
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    Power Series Approximation

    Use a power series to approximate
    <br />
\int cos(4x)ln(x) dx.<br />
    to six decimal places.

    Bounds are from pi to 2pi, if you care, but I'm not asking for you to solve the problem completely; just give me the steps.

    Thank you so much!
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  2. #2
    Member Abu-Khalil's Avatar
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    \int\cos 4x\log x dx = \frac{1}{4}\sin 4x\log x - \frac{1}{4}\int \frac{\sin 4x}{x}dx and use that \sin x = \sum_{k=0}^\infty(-1)^k\frac{(4x)^{2k+1}}{(2k+1)!}\Rightarrow \frac{\sin x}{x} = \sum_{k=0}^\infty(-1)^k\frac{4^{2k+1}x^{2k}}{(2k+1)!}
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