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Complicated integral

  1. Apr 17, 2014 #1

    wel

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    Gold Member

    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.
     
  2. jcsd
  3. Apr 17, 2014 #2

    CAF123

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    Gold Member

    Notice that you are only to consider the case when x is very small, tending to zero. This means you can make a suitable expansion of the exponential function, leaving a much simpler integral.
     
  4. Apr 18, 2014 #3
    Or maybe do this:
    $$I=I(0)+I'(0)x+O(x^2)$$
    :rolleyes:
     
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