$\int_{- 4}^{4} | x + 2 | d x$ is equal to :

50

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24

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20

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None of these.

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Solution

The correct option is D $20$
Let $f (x) = | x + 2 |$

Then, $f (x) = ⎧ ⎨ ⎩ \begin{matrix} (x + 2) & , i f x > - 2 - (x + 2) & , i f x < - 2 0 & , i f x = - 2 \end{matrix}$

$∴ \int_{- 4}^{4} | x + 2 | d x = \int_{- 4}^{- 2} - (x + 2) d x + \int_{- 2}^{4} (x + 2) d x$

$= {∣ ∣ ∣ \begin{matrix} \frac{- x^{2}}{2} - 2 x \end{matrix} ∣ ∣ ∣}_{- 4}^{- 2} + {∣ ∣ ∣ \begin{matrix} \frac{x^{2}}{2} + 2 x \end{matrix} ∣ ∣ ∣}_{- 2}^{4}$

$= - (2 - 8) - 2 (- 2 + 4) + (8 - 2) + 2 (4 + 2)$

$= 20$

Hence, option C is correct.

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