Question

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

Answer 1

Let ${\int }_2^8|x-5|dx.$
It can be seen that$(x-5)\le 0on[2,5]and(x-5)\ge 0on[5,8].\because {\int }_a^{b}f\left(x\right)dx={\int }_a^{c}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx\therefore I={\int }_2^5\{-(x-5\left)\right\}dx+{\int }_5^8(x-5)dx={[5x-\frac{x^2}{2}]}_2^5+{[\frac{x^2}{2}-5x]}_5^8=(25-\frac{25}{2})-(10-\frac{4}{2})+(\frac{64}{2}-40)-(\frac{25}{2}-25)=\frac{25}{2}-8-8+\frac{25}{2}=25-16=9$