Question

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

Answer 1

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