Evaluate the following integrals: ^ 6_ -6 |x+2 |dx - Vidyadip

Question

Evaluate the following integrals:$\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$

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Answer

$\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$We know that,
$|\text{x}+2|=\begin{cases}-(\text{x}+2),&-6\leq\text{x}\leq-2\\\text{x}+2,&-2<\text{x}\leq6\end{cases}$
$\therefore\ \text{I}=\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$
$\Rightarrow\text{I}=\int^\limits{-2}_{-6}\big(\text{x}+2\big)\text{dx}+\int^\limits6_{-2}\big(\text{x}+2\big)\text{dx}$
$\Rightarrow\text{I}=\Big[\frac{-\text{x}^2}{2}-2\text{x}\Big]^{-2}_{-6}+\Big[\frac{\text{x}^2}{2}-2\text{x}\Big]^6_{-2}$
$\Rightarrow\text{I}=-2+4+18-12+18+12-2+4$
$\Rightarrow\text{I}=40$

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