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Math Help - Aren't derivative and integral inverse operations?

  1. #1
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    Aren't derivative and integral inverse operations?

    Hey what's going on here. I thought if you take the derivative of a function, then integrate that derivative, you get the original function back.

    But when I perform this routine on the function below I get a different function from the original. I'd appreciate it if someone could point out where I've gone astray. Thanks.

    f(x)=\frac{x^2}{x^2+1}
    \frac{df}{dx}=\frac{2x}{(x^2+1)^2}
    \int\frac{df}{dx}dx=\frac{-1}{x^2+1}
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  2. #2
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    They are inverses - up to a constant. Your f(x) and the result of your integration differ by a constant (1, in this case). That is, if you add 1 to the result of your integration, you will get the function with which you started.
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