How can i find domain of this integral? $$\int\arccos\left(\sqrt{\frac{x-4}{x+6}}\right)dx$$
I tried this:
$$\frac{x-4}{x+6}\ge0\implies (-\infty,-6)\cup [4,\infty)$$ Next:
$$-1\le\sqrt{\frac{x-4}{x+6}}\le1$$ Can I do something like that? $-1\le\sqrt{\frac{x-4}{x+6}}$ $\cap\sqrt{\frac{x-4}{x+6}}\le1$
From the first constraint you have $$ \frac{x-4}{x+6} \ge 0 $$ and from the second one, $$ \left| \frac{x-4}{x+6} \right| \le 1 \Leftrightarrow -1 \le \frac{x-4}{x+6} \le 1 $$ and you need the intersection of both, yielding $$ 0 \le \frac{x-4}{x+6} \le 1. $$
Can you solve this?