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Math Help - Finding a function (f) and a number a that satisfy the definite integral

  1. #1
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    Finding a function (f) and a number a that satisfy the definite integral

    Question: Find a function (f) and a number 'a' that satisfy the definite integral
    Here is the integral: 6+∫[a,x,f(t)/t^2,t]=2sqrt(x) which comes out to

    Can somebody please explain how to do this? i tried using the FToC parts 1 and 2, but i think im using them wrong here...
    Thanks in advance!
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  2. #2
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    Re: Finding a function (f) and a number a that satisfy the definite integral

    Assume f is continuous. Then the derivative of \int_a^x {f(t)\over t^2}dt is {f(x)\over x^2}, which must then be {1\over \sqrt{x}} -- I took the derivative of both sides of the equation. So f(x)={x^2\over \sqrt{x}}=x^{3/2}. Now \int_a^x {t^{3/2}\over t^2}dt=\int_a^x t^{-1/2}dt=2t^{1/2}|_a^x=2\sqrt{x}-2\sqrt{a}. Finally then 6+2\sqrt{x}-2\sqrt{a}=2\sqrt{x} implies 3=\sqrt{a} or a=9.
    Thanks from topsquark
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