# Why does $\int \sec(x)dx=\ln|\tan(x)+\sec(x)| + C$? [duplicate]

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.

• – randomgirl May 27 at 18:04
• Think about integrals of the form $f'/f$ that result in logs. Now try to work back as suggested in answer. – Karl May 27 at 18:04
• I think common means like popular or well known here. – randomgirl May 27 at 18:05
• 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. – sirous May 27 at 19:11
• @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... – user209663 May 27 at 19:44

Hint: Multiply numerator and denominator of $$\sec(x)$$ by $$\tan(x)+\sec(x)$$