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I am trying to figure out why $\int \sec(x)dx=\ln|\tan(x)+\sec(x)| + C$. When I plugged it into symbolab, all it said was,

Use the common integral: $\int \sec(x)dx=\ln|\tan(x)+\sec(x)|$

I don't know what it means by "common integral". What is a "common integral"? I know what an integral is, but I couldn't find a definition of "common integral" specifically.

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  • $\begingroup$ Here is more intmath.com/methods-integration/table-common-integrals.php $\endgroup$ – randomgirl May 27 at 18:04
  • $\begingroup$ Think about integrals of the form $f'/f$ that result in logs. Now try to work back as suggested in answer. $\endgroup$ – Karl May 27 at 18:04
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    $\begingroup$ I think common means like popular or well known here. $\endgroup$ – randomgirl May 27 at 18:05
  • $\begingroup$ But If you take derivative of $Ln|tan(x)+Sec(x)|$, the result is not $Sec(x)$.Interested to where this result is come from. $\endgroup$ – sirous May 27 at 19:11
  • $\begingroup$ @sirous The derivative may not look like $\sec(x)$ at first but after some simplification you will see that it is... also you must remember that there is an extra constant after you integrated it... $\endgroup$ – user209663 May 27 at 19:44
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Hint: Multiply numerator and denominator of $$\sec(x)$$ by $$\tan(x)+\sec(x)$$

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