As per my understanding integral of $\frac{u'}{u}$ is $\ln|u|$

I tried doing that for evaluating the indefinite integral $\ln|sec(x)|$ So I got:

$$ \int \ln|sec(x)| \, dx = \frac{sec(x) \tan(x)}{ sec(x) } = \tan(x)$$

But that wasn't the right answer.

  1. So what would be the right answer and the logic behind it ?
  2. How I would find the integral of $\ln|\sin(x)|$ or $\ln|csc(x)|$ ?
  • $\begingroup$ Try integration by parts $\endgroup$ – Sak Feb 29 '16 at 4:56
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    $\begingroup$ The integral of $f'/f$ equals $\ln |f|$, not the other way around. $\endgroup$ – Mattos Feb 29 '16 at 4:59
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    $\begingroup$ According to W|A, there are no closed forms, but your function can be written in terms of polylogarithms. $\endgroup$ – Henricus V. Feb 29 '16 at 5:03
  • $\begingroup$ You just differentiated it. $\endgroup$ – Nikunj Feb 29 '16 at 10:28

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