Evaluate the following integrals: ^0_ -5 f(x)dx, Where f(x)=|x|+|… - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^0_{-5}\text{f(x)}\text{dx,}$ Where $\text{f(x)}=|\text{x}|+|\text{x}+2|+|\text{x}+5|$

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Answer

We have,
$\text{I}=\int\limits^0_{-5}\big(|\text{x}|+|\text{x}+2|+|\text{x}+5|\big)\text{dx}\\=\int\limits^0_{-5}|\text{x}|\text{dx}+\int\limits^0_{-5}|\text{x}+2|\text{dx}+\int\limits^0_{-5}|\text{x}+5|\text{dx}$
$\Rightarrow\text{I}=\int\limits^0_{-5}|\text{x}|\text{dx}+\int\limits^0_{-5}|\text{x}+2|\text{dx}+\int\limits^0_{-5}|\text{x}+5|\text{dx}$
$=\Big[\frac{-\text{x}^2}{2}\Big]^0_{-5}+\Big[\frac{-\text{x}^2}{2}-2\text{x}\Big]^{-2}_{-5}+\Big[\frac{\text{x}^2}{2}+2\text{x}\Big]^0_{-2}+\Big[\frac{\text{x}^2}{2}+5\text{x}\Big]^0_{-5}$
$=\Big[\frac{25}{2}\Big]-\big[\frac{4}{2}-4-\frac{25}{2}+10\Big]+\Big[0+0-\frac{4}{2}+4\Big]+\Big[0+0-\frac{25}{2}+25\big]$
$=\frac{25}{2}-\Big[8-\frac{25}{2}\Big]+\big[2\big]+\Big[25-\frac{25}{2}\Big]$
$=\frac{25}{2}-8+\frac{25}{2}+2+25-\frac{25}{2}$
$=19+\frac{25}{2}$
$=31\frac{1}{2}$
$=\frac{36}{2}$

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