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Thread: Integral of a function

  1. #1
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    Integral of a function

    I just want to see if this is right. I'm not sure I went down the right path.

    Find function f(x) and constant a such that


    I set x=a and found that a=3. Then I used the fundamental theorem of calculus to get rid of the integral. That left me with


    So my final answer is f(x) = x^3 - 3, and a=3. Is this right?
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  2. #2
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    Quote Originally Posted by ban26ana View Post
    I just want to see if this is right. I'm not sure I went down the right path.

    Find function f(x) and constant a such that


    I set x=a and found that a=3. Then I used the fundamental theorem of calculus to get rid of the integral. That left me with $\displaystyle \leftarrow$ should be $\displaystyle \frac{f(x)}{x^2} = 2$


    So my final answer is f(x) = x^3 - 3, and a=3. Is this right?
    . . .
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  3. #3
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    Thanks. Can I ask why that is? I'm a little more confused now.
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  4. #4
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    to make it easy, let $\displaystyle g(t) = \frac{f(t)}{t^2}$ and we have $\displaystyle 6 + \int_a^x g(t) ~dt = 2x$

    differentiate both side and using FTC, we have

    $\displaystyle \frac{d}{dx} \left[6 + \int_a^x g(t) ~dt \right]= \frac{d}{dx}\left[2x\right] \rightarrow g(x) = 2$
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