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Function involving definite integral

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Let g be a continuous function on R that satisfies ##\displaystyle g(x)+2\int_{0}^{\pi/2} \sin x \cos t g(t)dt=\sin x##, then ##g'\left(\frac{\pi}{3}\right)## is equal to
    A)1/2
    B)1/√2
    C)1/4
    D)none of these


    2. Relevant equations



    3. The attempt at a solution
    Rewriting the given expression,
    [tex]g(x)=\sin x\left(1-2\int_0^{\pi/2}\cos t g(t)dt \right)[/tex]
    [tex]g(x)=k\sin x[/tex]

    where ##\displaystyle k=\left(1-2\int_0^{\pi/2}\cos t g(t)dt \right)##.
    [tex]g'\left(\frac{\pi}{3}\right)=\frac{k}{2}[/tex]

    I am stuck here, how would I evaluate k?

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Apr 11, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You know that g(t)= k sin(t) so you can do the integral:
    [tex]k= 1- 2k\int_0^{2\pi} cos(t)sin(t) dt[/tex]
     
  4. Apr 11, 2013 #3
    Thanks HallsofIvy! Silly me, missed such an obvious step. :tongue2:
     
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