Evaluate the following integrals: ^2_ -2 |2x+3|dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^2_{-2}|2\text{x}+3|\text{dx}$

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Answer

$\int^\limits2_{-2}|2\text{x}+3|\text{dx}$
We know that,
$|2\text{x}+3|=\begin{cases}-(2\text{x}+3),&-2\leq\text{x}\leq-\frac{3}{2}\$2\text{x}+3),&-\frac{3}{2}<\text{x}\leq2\end{cases}$
$\therefore\ \text{I}=\int^\limits{\frac{-3}{2}}_{-2}-\big(2\text{x}+3\big)\text{dx}+\int^\limits2_{-\frac{3}{2}}\big(2\text{x}+3\big)\text{dx}$
$\Rightarrow\text{I}=-\Big[\text{x}^3+3\text{x}\Big]^{\frac{-3}{2}}_{-2}+\Big[\text{x}^2+3\text{x}\Big]^2_{-\frac{3}{2}}$
$\Rightarrow\text{I}=-\frac{9}{4}+\frac{9}{2}+4-6+4+6-\frac{9}{4}+\frac{9}{2}$
$\Rightarrow\text{I}=\frac{25}{2}$

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