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It seems to be true that:

$$\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$$

For eg., this works with $\frac{dF}{dt}=\frac{1}{2} (\cos(\pi \ln{t})+1)$

But then there must be something wrong because this would imply that $\int t dF=\int dF$ which seems unlikely.

I would appreciate it if someone could demonstrate why the first displayed equation is correct or incorrect.

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A counter-example to the above formula is $F(t) = \ln t$. Then $$ \int t \frac{d F}{d \ln t} d(\ln t) = \int dt = t, $$ but $$ \int \frac{dF}{dt}dt = \int \frac{1}{t}dt = \ln t. $$

You can perform the u-substitution $u = \ln t$ to see that that $$ \int t \frac{d F}{d \ln t} d(\ln t) $$

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