Value of ∫_-4^4 |x+2| dx is :

The correct answer is Option (1) → 20

The expression x + 2 changes sign at x = −2, so the modulus is removed separately on each side of it:

    \(|x+2| = -(x+2)\) for \(-4 \le x \le -2\)
    \(|x+2| = x+2\) for \(-2 \le x \le 4\)

So

    \( \displaystyle \int_{-4}^{4}|x+2|\,dx = \int_{-4}^{-2}-(x+2)\,dx + \int_{-2}^{4}(x+2)\,dx \)

    \( \displaystyle = \left[-\frac{(x+2)^2}{2}\right]_{-4}^{-2} + \left[\frac{(x+2)^2}{2}\right]_{-2}^{4} \)

    \( \displaystyle = \left(0 + \frac{4}{2}\right) + \left(\frac{36}{2} - 0\right) \)

    \( = 2 + 18 = 20 \)