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Math Help - Power Series for Indefinite Integral

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    Power Series for Indefinite Integral

    Find a power series for the indefinite integral of cos(x^2) dx. Use both sigma notation and expanded form, including the first four nonzero terms of the series.
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    Quote Originally Posted by leilani13 View Post
    Find a power series for the indefinite integral of cos(x^2) dx. Use both sigma notation and expanded form, including the first four nonzero terms of the series.
    \cos{X} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^nX^{2n}}{(2n)!}\right].


    So \cos{(x^2)} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^n(x^2)^{2n}}{(2n)!}\right]

     = \sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right].


    So now find

    \int{\cos{(x^2)}\,dx} = \int{\sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right]\,dx}.
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