Question

Solve :
${\int }_{-5}^5\left|x+2\right|dx$

Answer 1

${\int }_{-5}^5|x+2|dx$

Graph of $|x+2|$ is plotted as shown below.

${\int }_{-5}^5|x+2|dx$ means the area enclosed by $|x+2|$ between $-5$ and $5$.

$\therefore$ Given in triangle can be write $={\int }_{-5}^{-2}(-x+2)dx+{\int }_{-2}^5(x+2)dx$

$={[\frac{-x^2}{2}+2x]}_{-2}^{-5}+{[\frac{x^2}{2}+2x]}_{-2}^{-5}$

$=-4-\frac{4}{2}+\frac{25}{2}+10+\frac{25}{2}+10-\frac{4}{2}+4$

$=-4+25+20$

$=41$ (we have integral the line $-x+2$ from $-5$ to $-2$ to get area $\left(II\right)$ and line $x+2$ from $-2$ to $5$ to get area $\left(II\right)$)"