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Fundamental Theorem of Calculus to find Derivative

  1. Aug 5, 2009 #1
    Use the Fundamental Theorem of Calculus to find the derivative of the function



    g(x) = [tex]\sqrt{x}\int sinx[/tex] Ln(t) [tex]\frac{cos(t)}{t}[/tex] dt



    g'(x) = lnx cosx / x. By integrating this function, you receive the function g(x). Then by differentiating g(x) you receive g'(x) which is what is given, according to FTCI.


    I was told I got this completely wrong. (out of 5) So looking for some help. Thanks in advance.


    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 5, 2009 #2
    I do not understand something. Could you possibly rewrite the original problem?
    Is it:

    [tex]g(x)=\sqrt{x}\int{sin(x)dx} *\int{\frac{ln(t)*cos(t)}{t}dt} [/tex]


    Regards.
     
  4. Aug 5, 2009 #3

    jgens

    User Avatar
    Gold Member

    As stated, the function [tex]g(x)[/tex] doesn't look quite right. Is it supposed to be defined such that,

    [tex]g(x) = \sqrt x sin(x) \int_{a}^x \frac{ln(t)cos(t)}{t} \, dt[/tex]

    If so, then use the product rule of differentiation because [tex]g(x)[/tex] can be defined in terms of the product of two functions [tex]u[/tex] and [tex]v[/tex] where,

    [tex]u = \sqrt x sin(x)[/tex]

    [tex]v = \int_{a}^x \frac{ln(t)cos(t)}{t} \, dt[/tex]
     
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