A strange derivative-integral discrepancy

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In summary, the conversation discusses a discrepancy between the derivative and integral of the function ##\frac{x}{x+a}##. One person used Mathematica and got ##-\frac{a}{x+a}## as the integral, while another used a phone app and got ##\frac{x}{x+a}##. It is believed that the phone app is correct because it follows the principle of including a constant of integration, while the computer programs may have neglected this. The conversation ends with the realization that operations involving a common denominator can change the appearance of the function.
  • #1
ELB27
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Hi,
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.
 

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  • #2
very odd.
i used maxima (freeware mathematica):
integrate(a/(a^2+x^2+2*a*x),x)
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 don't understand why a phone app would beat 2 computer algebra systems though. !
 
  • #3
Let the constant of integration be 1. Then it's exactly the same.
 
  • #4
MarneMath said:
Let the constant of integration be 1. Then it's exactly the same.
Ah that's it - thanks. Interesting though how tricky these things can be.

DivergentSpectrum said:
very odd.
i used maxima (freeware mathematica):
integrate(a/(a^2+x^2+2*a*x),x)
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 don't understand why a phone app would beat 2 computer algebra systems though. !
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...
 

What is a strange derivative-integral discrepancy?

A strange derivative-integral discrepancy refers to a mathematical phenomenon where the derivative and integral of a function do not match. In other words, the derivative and integral of a function do not cancel each other out, as expected. This can occur due to various reasons, such as improper integration techniques or undefined points in the function.

Why is a strange derivative-integral discrepancy important?

A strange derivative-integral discrepancy is important because it highlights the limitations and complexities of mathematical concepts. It also serves as a reminder to carefully check the steps and assumptions made while solving a problem, as a small mistake can result in a significant discrepancy between the derivative and integral.

What are some examples of a strange derivative-integral discrepancy?

One common example is the function f(x) = |x|. The derivative of this function is f'(x) = 1 for x > 0 and f'(x) = -1 for x < 0. However, when integrated, the result is not the original function, but rather f(x) = x for x > 0 and f(x) = -x for x < 0. This discrepancy occurs due to the absolute value function, which is not differentiable at x = 0.

How can a strange derivative-integral discrepancy be resolved?

A strange derivative-integral discrepancy can be resolved by carefully checking the steps and assumptions made while solving the problem. It is also important to use proper integration techniques and consider the domain of the function. Sometimes, the discrepancy may also occur due to a mistake in the calculation, which can be corrected by carefully reviewing the work.

What are the implications of a strange derivative-integral discrepancy in real-world applications?

In real-world applications, a strange derivative-integral discrepancy can lead to incorrect results and solutions. This can have serious consequences in fields such as engineering and physics, where precise mathematical calculations are crucial. It is important for scientists and mathematicians to be aware of this phenomenon and carefully consider it while solving problems.

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