integration - why $int sqrt{(sin x)^2}, mathrm{d}x = int |sin x| ,mathrm{d}x$

May be this is a stupid question but why $$\int \sqrt{(\sin x)^2} \,\mathrm{d}x = \int |\sin x| \,\mathrm{d}x$$ instead of $$\pm \int \sin x \,\mathrm{d}x$$

I think may be because it violates the rule that a function can't have more than 1 output for a single input, which brings me to my next question does the intgrand need to be a function?

Answer

The definition of the square root, $\sqrt{\,\cdot\,}$, in real numbers is a function that given a non-negative number $y$, we take the non-negative number $x$ such that $x^2 = y$. That's why we write $$x^2 = k \Rightarrow x = \pm\sqrt{k}$$ instead of $$x^2 = k \Rightarrow x = \sqrt{k}.$$

We have to put the $\pm$ sign because the square root itself only considers the positive answer.

Observation: If the answer of square root was the two numbers, then it couldn't be a function.