$\underset{1}{\overset{5}{\int }}\{|x-1|+|x-2|+|x-3|\}dx=$

$I=\underset{1}{\overset{5}{\int }}\{|x-1|+|x-2|+|x-3|\}dx$

$=\underset{1}{\overset{2}{\int }}[(x-1)+(2-x)+(3-x)]dx+\underset{2}{\overset{3}{\int }}[(x-1)+(x-2)+(3-x)]dx$

$+\underset{4}{\overset{5}{\int }}\left[((x-1)+(x-2)+(x-3))\right]dx$

Splitting rule :

$=\frac{12-7+5+15+9}{2}=\frac{41-7}{2}=\frac{34}{2}=17$

$|x-2|=\left\{\begin{array}{cc}x-2& x\ge 2\\ -(x-2)& x<2\end{array}\right.$

$|x-3|=\left\{\begin{array}{cc}x-3& x\ge 3\\ -(x-3)& x<3\end{array}\right.$

$=\underset{1}{\overset{2}{\int }}(4-x)dx+\underset{2}{\overset{3}{\int }}xdx+\underset{3}{\overset{5}{\int }}(3x-6)dx$

$={4x-\frac{x^2}{2}]}_1^2+{\frac{x^2}{2}]}_2^3+3{\left(\frac{{\displaystyle x^2}}{{\displaystyle 2}}-2x\right)}_3^5$

$=(8-2)-\left(4-\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)+\left(\frac{{\displaystyle 9}}{{\displaystyle 2}}-2\right)+3\left[\left(\frac{{\displaystyle 25}}{{\displaystyle 2}}-10\right)-\left(\frac{{\displaystyle 9}}{{\displaystyle 2}}-6\right)\right]$

$=6-\frac{7}{2}+\frac{5}{2}+\frac{15}{2}+\frac{9}{2}$

$=\frac{12-7+5+15+9}{2}=\frac{41-7}{2}=\frac{34}{2}=17$.