$$\int \tan x \, dx \equiv \ln|\sec x|$$
What's the significance of the modulus?
I remember doing a question using this formula, and had a negative answer inside the log, so I gave up - because I was using the log with normal brackets.
I know you can't log a negative, but that doesn't help me understand why this mod is here.
On
Two points I'd like to make:
Notice that $\sec x$ comes in intervals of being positive and negative.
Notice that $\tan x$ diverges at the endpoints of the above intervals, so whatever you do, you can't integrate over those points.
Now notice that these are trig functions, and thus periodic, so every integral you take reduces down to one interval (since every interval is the same), and so to preserve this, we take the absolute value of $\sec x$ to get a positive number to put into the integrand.
Real values integrate to real values, so if we didn't have the modulus symbol we would be taking the natural logarithm of a negative number when $\sec x < 0$. Hope it helps.