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In a problem, the substitution $$\tan\theta=\frac{x}{2}$$ was made. In the end, the answer was in terms of sines, and to convert back, $sin\theta$ was defined as $$\sin\theta=\frac{x}{\sqrt{4+x^2}}$$ This is a typical example of some stuff about integration I'm struggling to understand;

(1) Why are the absolute values of square roots never taken? This is something I keep seeing in every situation involving an integral. (Here, if $\theta$ is in the third quadrant, sines would be negative and tans would be positive. So this definitely doesn't work for the third quadrant.)

(2) Expanding upon the stuff in the parantheses up there, a possible explanation is that while doing trig substitutions, the angle is always a principal angle of the inverse trigonometric operation on whatever you're making the substitution. Is there such a rule?