Evaluate the following integrals: ^8_2|x-5|dx - Vidyadip

Question

Evaluate the following integrals:$\int\limits^8_2|\text{x}-5|\text{dx}$

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Answer

$\int\limits^8_2|\text{x}-5|\text{dx}$We know that,
$|\text{x}-5|=\begin{cases}-(\text{x}-5),&2\leq\text{x}\leq5\\\text{x}-5,&5<\text{x}\leq8\end{cases}$
$\therefore\ \text{I}=\int\limits^8_2|\text{x}-5|\text{dx}$
$\Rightarrow\text{I}=\int\limits^5_2-(\text{x}-5)\text{dx}+\int\limits^8_5(\text{x}-5)\text{dx}$
$\Rightarrow\text{I}=-\Big[\frac{\text{x}^2}{2}-5\text{x}\big]^5_2+\Big[\frac{}{}\frac{\text{x}^2}{2}-5\text{x}\Big]^8_5$
$\Rightarrow\text{I}=\frac{-25}{2}+25+2-10+32-40-\frac{25}{2}+25$
$\Rightarrow\text{I}=9$

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