Mathematics - Definite Integration Question with Solution | TestHub

$f\left(x\right)=$ max $\left\{\left|x+1\right|,\left|x+2\right|,\left|x+3\right|,\left|x+4\right|,\left|x+5\right|\right\}$

${\int }_{-6}^{0}f\left(x\right)dx={\int }_{-6}^{-3}\left|x+1\right|dx+{\int }_{-3}^0\left|x+5\right|dx$

$=-{\int }_{-6}^{-3}\left(x+1\right)dx+{\int }_{-3}^0\left(x+5\right)dx$

$=-{\left[\frac{x^2}{2}+x\right]}_{-6}^{-3}+{\left[\frac{x^2}{2}+5x\right]}_{-3}^0$

$=-\left[\left(\frac{9}{2}-3\right)-\left(18-6\right)\right]+\left[0-\left(\frac{9}{2}-15\right)\right]$

$=-\left[\frac{3}{2}-12\right.⌉+\frac{21}{2}=\frac{21}{2}+\frac{21}{2}=21$