It's a common technique, when presented with (for example) $\displaystyle \int \! \sqrt{u^2+1}\,\mathrm d u$, to make a substitution such as $\displaystyle u=\tan{\theta}$ and then we have that the previous integral is equal to $\displaystyle \int \! \sqrt{\tan^2{\theta}+1}\sec^2{\theta}\, \mathrm d \theta=\int \! \sqrt{\sec^2{\theta}}\sec^2{\theta}\, \mathrm d \theta$. Here is the point I'm concerned about - what allows us to proceed to $\displaystyle \int \! \sec^3{\theta}\, \mathrm d \theta$, as is the common practice? Our limits may not be such that $\displaystyle sec{\theta}>0$, so what justifies simplifying the square root portion?