Evaluate the following integrals: ^ 3_ -3 |x+1|dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits3_{-3}|\text{x}+1|\text{dx}$

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Answer

$\text{I}=\int^\limits3_{-3}|\text{x}+1|\text{dx}$
We know that,
$|\text{x}+1|=\begin{cases}-(\text{x}+1),&-3\leq\text{x}\leq-1\\\text{x}+1,&-1<\text{x}\leq3\end{cases}$
$\therefore\ \text{I}=\int^\limits{-1}_{-3}-(\text{x}+1)\text{dx}+\int^\limits{3}_{-1}\big[\text{x}+1\big]\text{dx}$
$\Rightarrow\text{I}=\bigg[-\frac{(\text{x}+1)^2}{2}\bigg]+\bigg[\frac{(\text{x}+1)^2}{2}\bigg]^3_{-1}$
$\Rightarrow\text{I}=0+2+8-0$
$\Rightarrow\text{I}=10$

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