Question

Evaluate: $\int {}_{-5}^3\left|x+2\right|dx$

Answer 1

Solution :- given

${\Rightarrow }_{-5}{\int }^3|x+2|dx$

for $-5\le x\le -2:(-x-2)$

$-2\le x\le 3:(x+2)$

$\therefore$ Integration can be written as

${=}_{-5}{\int }^{-2}(-x-2)dx{+}_{-2}{\int }^3(x+2)dx$

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

$=\left[\right(\frac{-(-2)}{2}+2\times 2)-(\frac{(-5{)}^2}{2})+2\times 5]+\left[\right(\frac{3^2}{2}+2.3)-(\frac{(-2{)}^2}{2}+2.(-2)\left)\right]$

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

$=\left[\right(-2+4)-(\frac{-25+20}{2}\left)\right]+\left[\right(\frac{9+12}{2})-(2-4\left)\right]$

$=[2+\frac{5}{2}]+[\frac{21}{2}+2]$

$=\frac{9}{2}+\frac{25}{2}$

$=\frac{34}{2}$

$=17$ Ans