Evaluate the following integrals: ^2_ -2 |x+1|dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits{2}_{-2}|\text{x}+1|\text{dx}$

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Answer

$\int^\limits{2}_{-2}|\text{x}+1|\text{dx}=\int^\limits{-1}_{-2}-(\text{x}+1)\text{dx}+\int^\limits{2}_{-1}(\text{x}+1)\text{dx}$
$=-\Big[\frac{\text{x}^2}{2}+\text{x}\Big]^{-1}_{-2}+\Big[\frac{\text{x}^2}{2}+\text{x}\Big]^2_{-1}$
$=-\bigg[\Big(\frac{1}{2}-1\Big)-\Big(\frac{4}{2}-2\Big)\bigg]+\bigg[\Big(\frac{4}{2}-2\Big)-\Big(\frac{1}{2}-1\Big)\bigg]$
$=-\bigg[\Big(-\frac{1}{2}\Big)-0\bigg]+\Big[4+\frac{1}{2}\Big]$
$=\frac{1}{2}+4\frac{1}{2}$
$=5$

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