By Using properties of definite integral, evaluate the following … - Vidyadip

Question

By Using properties of definite integral, evaluate the following integral in Exercise:
$\int^{4}_{0}|\text{x}-1|\text{dx}$

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Answer

$\text{Let}\ \text{I}\int^{4}\limits_{0}|\text{x}-1|\text{dx}$
$\text{Here}\ \ \text{x}-1=0\ \ \Rightarrow\ \ \text{x}=1\in(0,4)$
$\therefore\ \ \text{from eq. (i)},\text{I}=\int^{1}\limits_{0}|\text{x}-1|\text{dx}+\int^{4}\limits_{1}|\text{x}-1|\text{dx}=-\int^{1}\limits_{0}(\text{x}-1)\text{dx}+\int^{4}\limits_{1}(\text{x}-1)\text{dx}$
$\Rightarrow\ \ \text{I}=-\bigg(\frac{\text{x}^{2}}{2}-\text{x}\bigg)^1_0+\bigg(\frac{\text{x}^{2}}{2}-\text{x}\bigg)^{4}_{1}$
$=-\Big\{\frac{1}{2}-1-0\Big\}+\Big\{\frac{16}{2}-4-\frac{1}{2}+1\Big\}$
$\Rightarrow\ \ \text{I}=\frac{-1}{2}+1+8-4-\frac{1}{2}+1=6-\frac{2}{2}=6-1=5$

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