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Math Help - Integration

  1. #1
    Junior Member
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    Integration

    Consider the integral
    \begin{equation}
    I(x)=\int^{2}_{0} (1+t) e^{xcos[\pi (t-1)/2]} dt
    \end{equation}
    show that
    \begin{equation}
    I(x)= 4+ \frac{8}{\pi}x +O(x^{2})
    \end{equation}
    as $x\rightarrow0$.


    => Using integration by parts, but its too complicated for me because of huge exponential term.
    please help me.
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  2. #2
    MHF Contributor
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    Re: Integration

    Expand the exponential term as a taylor series about x=0.

    Keep the terms below 2nd order in t (the first 2 terms)

    Plug that into your integral and solve.
    Thanks from prasum and grandy
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  3. #3
    MHF Contributor
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    Re: Integration

    Another way

    I(x) = I(0) + I'(0)x + O(x^2)

    Then calculate I(0) and I'(0).
    Thanks from prasum and grandy
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