Evaluate the following integrals: ^2_ 1 |x-3|dx - Vidyadip

Question

Evaluate the following integrals:$\int^\limits{2}_{1}|\text{x}-3|\text{dx}$

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Answer

$\int^\limits{2}_{1}|\text{x}-3|\text{dx}$We know that,
$|\text{x}+1|=\begin{cases}-(\text{x}+1),&1\leq\text{x}\leq3\$\text{x}+1),&\text{x}>3\end{cases}$
$\therefore\ \text{I}=\int^\limits{2}_{1}|\text{x}-3|\text{dx}$
$\Rightarrow\text{I}=\int^\limits{2}_{1}-(\text{x}-3)\text{dx}$
$\Rightarrow\text{I}=\Big[\frac{-\text{x}^2}{2}-3\text{x}\Big]^2_1$
$\Rightarrow\text{I}=-2-6+\frac{1}{2}+3$
$\Rightarrow\text{I}=\frac{3}{2}$

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