If I have this expression:$$\exp\left(\ln\left|\sec(x)\right|+m\right)$$
can I rewrite it as: $$\sec(x)\exp(m)$$
with no problems? I'm making the assumption that I can drop the absolute value signs since they were only necessary to avoid negative values in the natural log function.
I'm asking because later I end up with this expression in an integral like this: $$\int \sin(2x)\left|\sec(x)\right|dx$$ and I'd like to be able to use $\sin(2x) = 2\sin(x)\cos(x)$ to cancel the $\cos(x)$ functions so that I can get $$\int2\sin(x)dx$$ If I drop the absolute value signs it cancels no problem. Is this allowed? Or should I use u-substitution with $u = \cos(x)$?