#### SamRoss

Gold Member

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- Summary
- Confusion with cancelling two dx "terms" as if they were simple fractions.

In Paul Nahin's book

## \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.

__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.