Answer
$$6$$
Work Step by Step
\begin{align*}
\int_{-1}^{4} \frac{d x}{\sqrt{|x|}}&=\lim _{b \rightarrow 0^{-}} \int_{-1}^{b} \frac{d x}{\sqrt{-x}}+\lim _{c \rightarrow 0^{+}} \int_{c}^{4} \frac{d x}{\sqrt{x}}\\
&=\lim _{b \rightarrow 0^{-}}[-2 \sqrt{-x}]\bigg|_{-1}^{b}+\lim _{c \rightarrow 0^{+}}[2 \sqrt{x}]\bigg|_{c}^{4}\\
&=\lim _{b \rightarrow 0^{-}}[(-2 \sqrt{-b})-(-2 \sqrt{-(-1)})]+\lim _{c \rightarrow 0^{+}}[2 \sqrt{4}-2 \sqrt{c}]\\
&=0+2+2 \cdot 2-0=6
\end{align*}
