By using the properties of definite integral, evaluate the integr… - Vidyadip

Question

By using the properties of definite integral, evaluate the integral in Exercise:
$\int\limits_{2}^{8}|\text{x}-5|\ \text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{2}^{8}|\text{x}-5|\ \text{dx}$

$\text{putting}\ \text{x}-5=0\ \ \Rightarrow\ \text{x}=5\in(2,8)$

$\therefore\ \ \text{from eq. (i)},\ \text{I}=\int\limits_{2}^{5}|\text{x}-5|\ \text{dx}+\int\limits_{5}^{8}|\text{x}-5| \text{dx}$

$=\int\limits_{2}^{5}-(\text{x}-5)\ \text{dx}+\int\limits_{5}^{8}(\text{x}-5)\text{dx}$

$=-\bigg(\frac{\text{x}^{2}}{2}-5\text{x}\bigg)^{-2}_{-5}+\bigg(\frac{\text{x}^{2}}{2}-5\text{x}\bigg)^{5}_{-2}$

$=-\bigg[\bigg(\frac{25}{2}-25\bigg)-(10-2)\bigg]+\bigg[(32-40)-\bigg(\frac{25}{2}-25\bigg)\bigg]$

$=25-\frac{25}{2}-8-8-\frac{25}{2}+25=34-\frac{50}{2}=34-25=9$

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