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Homework Help: Finding an Integral from an Unknown Function

  1. Dec 2, 2008 #1
    1. The problem statement, all variables and given/known data
    If xsin[tex]\pi[/tex]x = [tex]\int[/tex]f(t) dt, where is a continous function, find f(4).
    b=x2, a=0

    2. Relevant equations



    3. The attempt at a solution
    I assumed that the problem dealt with the Fundamental Theroem of Calculus so I began by saying that g(x)=xsin[tex]\pi[/tex]x but that is as far as it makes sense to me. Could anyone confirm to me that I am following the correct path?
     
  2. jcsd
  3. Dec 2, 2008 #2

    HallsofIvy

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    Science Advisor

    Yes, if
    [tex]x sin(\pi x)= \int_0^{x^2} f(t)dt[/quote]
    you can apply the Fundamental theorem but if you do that, because the upper limit is "x2" rather than "x", you need to use the chain rule also:
    [tex]F(x)= \int_0^x^2 f(t)dt= \int_0^u f(t)dt[/tex]
    where u= x2. The derivative of the right hand side is F'(u)(du/dx)= f(u)(2x)= f(x2)(2x). That is equal to the derivative of the left hand side. What is the derivative of [itex]x sin(\pi x)[/itex]?
    [tex]f(
     
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