Question

If [$x$] denotes the greatest integer less than or equal to $x$, then the value of $\underset{1}{\overset{5}{\int }}\left[|x-3|\right]dx$ is equal to

Answer 1

$\underset{1}{\overset{5}{\int }}\left[|x-3|\right]dx=\underset{1}{\overset{3}{\int }}[3-x]dx+\underset{3}{\overset{5}{\int }}[x-3]dx=\underset{1}{\overset{3}{\int }}[3-x]dx+\underset{3}{\overset{5}{\int }}[x-3]dx=\underset{1}{\overset{2}{\int }}[3-x]dx+\underset{2}{\overset{3}{\int }}[3-x]dx+\underset{3}{\overset{4}{\int }}[x-3]dx+\underset{4}{\overset{5}{\int }}[x-3]dx=\underset{1}{\overset{2}{\int }}1\cdot dx+\underset{2}{\overset{3}{\int }}0\cdot dx+\underset{3}{\overset{4}{\int }}0\cdot dx+\underset{4}{\overset{5}{\int }}1\cdot dx=2-1+5-4=2$