Question

By using the properties of definite integrals, evaluate the integral ${\int }_{-5}^5|x+2|dx$

Answer 1

Let $I={\int }_{-5}^5|x+2|dx$
It can be seen that $(x+2)\le 0$ on $[-5,-2]$ and $(x+2)\ge 0$ on $[-2,5]$.
$\therefore I={\int }_{-5}^{-2}-(x+2)dx+{\int }_{-2}^5(x+2)dx$
$=-{\left[\frac{x^2}{2}+2x\right]}_{-5}^{-2}+{\left[\frac{x^2}{2}+2x\right]}_{-2}^5$
$=-\left[\frac{(-2{)}^2}{2}+2(-2)-\frac{(-5{)}^2}{2}-2(-5)\right]+\left[\frac{(5{)}^2}{2}+2\left(5\right)-\frac{(-2{)}^2}{2}-2(-2)\right]$
$=-[2-4-\frac{25}{2}+10]+\left[\frac{25}{2}+10-2+4\right]$
$=-2+4+\frac{25}{2}-10+\frac{25}{2}+10-2+4=29$