Question

By using the properties of definite integrals, evaluate the integral ${\int }_2^8|x-5|dx$

Answer 1

Let $I={\int }_2^8|x-5|dx$
It can be seen that $(x-5)\le 0$ on $[2,5]$ and $(x-5)\ge 0$ on $[5,8]$.
$\therefore I={\int }_2^5-(x-5)dx+{\int }_1^8(x-5)dx$
$=-{[\frac{x^2}{2}-5x]}_2^5+{[\frac{x^2}{2}-5x]}_2^8$
$=-[\frac{25}{2}-25-2+10]+[32-40-\frac{25}{2}+25]$
$=9$