Justification for cancelling dx in an integral

In summary: The integral of an exact differential form is the difference of the values of the function, so it 'undoes' the differential.
  • #1
SamRoss
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TL;DR Summary
Confusion with cancelling two dx "terms" as if they were simple fractions.
In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))...

## \int_0^\phi \frac {d} {dx} f(x) dx = \int_0^\phi d{f(x)} = f(x) |_0^\phi##

It looks like the dx in the derivative symbol is cancelling with the dx from the integral, leaving only a ##d##. I have always been somewhat confused when derivatives are treated as fractions. How is what is being done here justified? Also, what is the meaning of ## df(x) ##? Finally, how does one get from the middle expression to the final expression? It looks like an example of integration and differentiation being inverses of each other but I'm not sure.
 
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  • #2
SamRoss said:
In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))...

## \int_0^\phi \frac {d} {dx} f(x) dx = \int_0^\phi d{f(x)} = f(x) |_0^\phi##

It looks like the dx in the derivative symbol is cancelling with the dx from the integral, leaving only a ##d##. I have always been somewhat confused when derivatives are treated as fractions. How is what is being done here justified? Also, what is the meaning of ## df(x) ##? Finally, how does one get from the middle expression to the final expression? It looks like an example of integration and differentiation being inverses of each other but I'm not sure.
A formal proof why this solppiness can be done is a bit of work to do as it involves two limits (Riemann integration and differential quotient).

The question about ##df(x)## is easier: Just set ##y=f(x)## then ##\int df(x) =\int dy = \int 1\cdot dy = y +C= f(x)+C##.
 
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  • #3
SamRoss said:
In Paul Nahin's book Inside Interesting Integrals, on pg. 113, he writes the following line (actually he wrote a more complicated function inside the integral where I have simply written f(x))...

## \int_0^\phi \frac {d} {dx} f(x) dx = \int_0^\phi d{f(x)} = f(x) |_0^\phi##

It looks like the dx in the derivative symbol is cancelling with the dx from the integral, leaving only a ##d##. I have always been somewhat confused when derivatives are treated as fractions. How is what is being done here justified? Also, what is the meaning of ## df(x) ##? Finally, how does one get from the middle expression to the final expression? It looks like an example of integration and differentiation being inverses of each other but I'm not sure.

Now that I think about it, does the second expression follow from the first simply by the definition of a differential, which I think is ## df = f'(x)dx ## ? In that case, can the second expression be skipped altogether so we can just go from the first to the last by the first fundamental theorem of calculus?
 
  • #4
SamRoss said:
Now that I think about it, does the second expression follow from the first simply by the definition of a differential, which I think is ## df = f'(x)dx ## ? In that case, can the second expression be skipped altogether so we can just go from the first to the last by the first fundamental theorem of calculus?
Yes, that's an idea, although it would only be a rewording.
 
  • #5
If you want to read a bit more, lookup differential forms, closed and exact forms. d is the differential operator; ( somewhat confusing), the differential of a differentiable function is a differential form.
 
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1. What is the reason for cancelling dx in an integral?

The reason for cancelling dx in an integral is to simplify the integral and make it easier to solve. It is a common technique used in integration to eliminate unnecessary terms and make the expression more manageable.

2. Is it always appropriate to cancel dx in an integral?

No, it is not always appropriate to cancel dx in an integral. This technique should only be used when the integral can be simplified and the resulting expression is still equivalent to the original integral.

3. Can cancelling dx change the value of the integral?

Yes, cancelling dx can change the value of the integral. This is why it is important to carefully consider when and how to use this technique, as it can affect the accuracy of the solution.

4. Are there any rules or guidelines for cancelling dx in an integral?

Yes, there are some rules and guidelines to follow when cancelling dx in an integral. These include ensuring that both the numerator and denominator of the expression being cancelled contain dx, and checking for any potential errors or changes in value.

5. How do I know when it is appropriate to cancel dx in an integral?

Knowing when to cancel dx in an integral comes with practice and experience. It is important to carefully analyze the integral and consider if cancelling dx will simplify the expression without changing its value. If in doubt, it is best to seek guidance from a teacher or textbook.

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