Problem: how to evaluate $\int_0^y[-\frac{d}{dx}(\ln(f(x))]dx$?

If we would have had $\int[-\frac{d}{dx}(\ln(f(x))]dx$ then the integration and differentiation would have killed each other but since we have a definite integral that's not possible.

Using Leibniz' Rule is possible but makes it more difficult.

In the derivation in my textbook they seem to ignore the derivative in its entirety and immediately apply the Fundamental Theorem of Calculus, i.e. they write $\int_0^y[-\frac{d}{dx}(\ln(f(x))]dx=-(\ln(f(y)-\ln(f(0))$.

I seem to be overlooking something. What is the correct approach and interpretation here?

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    $\begingroup$ On the contrary, Fundamental Theorem of Calculus involves exactly definite integrals and not indefinite. The proof in the book is correct. $\endgroup$ – wilkersmon Feb 14 '18 at 15:09
  • $\begingroup$ Whey happens when you perform the derivative first? $\endgroup$ – Michael McGovern Feb 14 '18 at 15:10
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    $\begingroup$ If you perform the derivative first, it's the second part of FToC, otherwise referred to as Newton–Leibniz axiom $\endgroup$ – wilkersmon Feb 14 '18 at 15:11
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    $\begingroup$ Informally, the fundamental theorem of calculus gives $\int_0^x g'(t)dt = g(x)-g(0)$. Here $g(t) = - \ln ( f(x))$. $\endgroup$ – copper.hat Feb 14 '18 at 15:24

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