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Integration by expansion

  1. Apr 18, 2014 #1

    wel

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    Gold Member

    Consider the integral
    \begin{equation}
    I(x)= \frac{1}{\pi} \int^{\pi}_{0} sin(xsint) dt
    \end{equation}
    show that
    \begin{equation}
    I(x)= 4+ \frac{2x}{\pi}x +O(x^{3})
    \end{equation}
    as [itex]x\rightarrow0[/itex].

    => I Have used the expansion of McLaurin series of [itex]I(x)[/itex] but did not work.
    please help me.

    (Note: It is not my homework or coursework question but it is from past exam paper which i am preparing for my exam)
     
  2. jcsd
  3. Apr 18, 2014 #2

    Simon Bridge

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    Science Advisor
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    Gold Member
    2016 Award

    Please show your working.

    Maclearen series: $$f(t)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$

    ##f(t)=\sin(x\sin t),\; f(0)=0##

    ##f'(t)=\cdots##

    ##f''(t)=\cdots##

    etc. until you start getting terms in x3
     
  4. Apr 19, 2014 #3

    Mark44

    Staff: Mentor

    wel,
    Physics Forums rules require that you show what you have tried. I sent you a PM about this, and am closing this thread. You are welcome to start a new thread for this problem, but you need to show what you have tried.
     
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