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Math Help - Calculus 1 Integrals FTC

  1. #1
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    Calculus 1 Integrals FTC

    Find a function f and a number "a" such that

    2 + int_(x at top, a at bottom) f(t)/t^3 dt = 6x^-3

    ?? I've been learning the fundamental theorem of calculus but I've just been staring at this question. I really don't know where to begin. I didn't think the lower limit, a, should come into play with the FTC... I don't know.

    Please help. Thanks!!
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    Even if someone could show me an example of another problem like this and how to solve it that would be helpful! I just don't know the process and I can't find a similar problem in my textbook or online.
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    Quote Originally Posted by littlejodo View Post
    Find a function f and a number "a" such that

    2 + int_(x at top, a at bottom) f(t)/t^3 dt = 6x^-3

    ?? I've been learning the fundamental theorem of calculus but I've just been staring at this question. I really don't know where to begin. I didn't think the lower limit, a, should come into play with the FTC... I don't know.

    Please help. Thanks!!
    2 + \int_x^a \frac{f(t)}{t^3} \, dt = 6x^{-3}

    2 - \int_a^x \frac{f(t)}{t^3} \, dt = 6x^{-3}

    \frac{d}{dx} \left[2 - \int_a^x \frac{f(t)}{t^3} \, dt = 6x^{-3}\right]

    -\frac{f(x)}{x^3} = -18x^{-4}

    f(x) = \frac{18}{x}
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    So you never have to take the constant 2 into consideration?

    And what does it mean to find an "a" value in this problem? It doesn't seem like the value of a mattered at all.
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  5. #5
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    Quote Originally Posted by littlejodo View Post
    So you never have to take the constant 2 into consideration?
    the derivative of a constant = 0

    And what does it mean to find an "a" value in this problem? It doesn't seem like the value of a mattered at all.
    it didn't when taking the derivative, but you can find the value of a now that you have f(x) ...

    2 - \int_a^{x} \frac{18}{t^4} \, dt = 6x^{-3}

    2 - \left[-\frac{6}{t^3}\right]_a^x = 6x^{-3}

    2 - \left[-\frac{6}{x^3} + \frac{6}{a^3}\right] = 6x^{-3}

    2 + \frac{6}{x^3} - \frac{6}{a^3} = 6x^{-3}

    a^3 = 3

    a = \sqrt[3]{3}
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