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A strange derivative-integral discrepancy

  1. Dec 4, 2014 #1
    I have just encountered a strange (for me) inconsistency with a derivative: If I take the x derivative of ##\frac{x}{x+a}## I get ##\frac{x+a-x}{(x+a)^2} = \frac{a}{(x+a)^2}##. However, when I take the integral of the latter I get: ##-\frac{a}{x+a} (+constant) ≠ \frac{x}{x+a}##. I have checked the above (simple) calculations with Mathematica to make sure I didn't make any mistake with formulas/arithmetic. Maybe I forget some principle but how is it possible to integrate the derivative of a function and get something else than the function itself back?
    Appreciate any help on this.
    Just for the record, I have attached a screenshot of Mathematica.

    Attached Files:

    Last edited: Dec 4, 2014
  2. jcsd
  3. Dec 4, 2014 #2
    very odd.
    i used maxima (freeware mathematica):
    and got -a/(x+a) like youre saying

    then i tried realcalc (a phone app with a symbolic integration feature) and i got x/(a+x)
    obviously realcalc is right, i dont understand why a phone app would beat 2 computer algebra systems though. !!!
  4. Dec 4, 2014 #3


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    Let the constant of integration be 1. Then it's exactly the same.
  5. Dec 4, 2014 #4
    Ah that's it - thanks. Interesting though how tricky these things can be.

    Maybe the app is programmed so that the integral of a derivative of anything would give the same thing back. As for the computer programs - they obviously neglected the constant of integration and took it to be 0.
    I wonder if all this operations of common denominator are legal given how they sneakily change the appearance of the function...
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