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

Question

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

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Answer

$\text{Let}\ \text{I}=\int\limits_{-5}^{5}|\text{x}+2|\ \text{dx}$
$\text{putting}\ \ \text{x}+2=0\ \ \Rightarrow=-2\in(-5,5)$
$\therefore\ \ \text{From eq. (i)},\ \text{I}=\int\limits_{-5}^{-2}|\text{x}+2|\ \text{dx}+\int\limits_{-2}^{5}|\text{x}+2|\ \text{dx}=\int\limits_{-5}^{-2}-\text{(x}+2 )\ \text{dx}+\int\limits_{-2}^{5}\text{(x}+2)\ \text{dx}$
$=-\bigg(\frac{\text{x}^{2}}{2}+2\text{x}\bigg)^{-2}_{-5}+\bigg(\frac{\text{x}^{2}}{2}+2\text{x}\bigg)^{5}_{-2}=-\bigg[\bigg(\frac{4}{2}-4\bigg)-\bigg(\frac{25}{2}-10\bigg)\bigg]+\bigg[\bigg(\frac{25}{2}-10\bigg)-\bigg(\frac{4}{2}-4\bigg)\bigg]$
$=-\bigg(-2-\frac{5}{2}\bigg)+\bigg(\frac{45}{2}+2\bigg)=2+\frac{5}{2}+\frac{45}{2}+2=4+25=29$

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