Evaluate: ^ 4 _ 1 |x - 1| + |x - 2| + |x - 4| dx - Vidyadip

Question

Evaluate:
$\int\limits^{4}_{1} \left\{|\text{x - 1}| + |\text{x - 2}| + |\text{x - 4}| \right\} \text{dx}$

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Answer

$\text{I} = \int\limits^{4}_{1} \left\{|\text{x - 1}| + |\text{x - 2}| + |\text{x - 4}| \right\} \text{dx}$
$= \int\limits^{4}_{1} \text{(x - 1) dx} - \int\limits^{2}_{1} \text{(x - 2)} \text{dx} + \int\limits^{4}_{2} \text{(x - 2) dx} - \int\limits^{4}_{1} \text{x - 4) dx}$
$= \bigg[\frac{\text{(x - 1)}^{2}}{2}\bigg]^{4}_{1} - \bigg[\frac{\text{(x - 2)}^{2}}{2}\bigg]^{2}_{1} + \bigg[ \frac{\text{(x - 2)}^{2}}{2}\bigg]^{4}_{2} - \bigg[\frac{\text{(x - 4)}^{2}}{2}\bigg]^{4}_{1}$
$= \frac{9}{2} + \frac{1}{2} +2+ \frac{9}{2} = 11 \frac{1}{2} \text{ or } \frac{23}{2}$

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