$$\int_{-2}^{8}\dfrac{dx}{\sqrt{|2x\|}}$$
I understand that you have to split this into two integrals because at $x=0$, the function is not defined. The example showed that they split up the integral like this:
$$\lim_{b\to0^{-}}\int_{-2}^{b}\dfrac{dx}{\sqrt{-2x}}+\lim_{c\to0^{+}}\int_{c}^{8}\dfrac{dx}{\sqrt{2x}}$$
I understand how they split up the integral but why is the denominator different in each one? I figured that the first integral has negative limits of integration so a negative has to be put in the square root to make it positive. The integral on the right hand side remains positive because the limits are never negative. Is this the right assumption?
If my assumption is correct, is there a general rule for dealing with absolute values in problems like these? I appreciate anyone that would take the time to explain this to me. Thank you in advance!
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