The value of ∫_-5^5 |x + 2| dx is

The correct answer is Option (1) → 29

The expression x + 2 changes sign at x = −2:

    \( |x + 2| = -(x + 2) \) for \( -5 \le x \le -2 \)
    \( |x + 2| = x + 2 \) for \( -2 \le x \le 5 \)

So the integral is split at −2:

    \( \displaystyle \int_{-5}^{-2}-(x + 2)\,dx = -\Big[\frac{(x + 2)^{2}}{2}\Big]_{-5}^{-2} \)

    \( \displaystyle = -\left(0 - \frac{9}{2}\right) = \frac{9}{2} \)

    \( \displaystyle \int_{-2}^{5}(x + 2)\,dx = \Big[\frac{(x + 2)^{2}}{2}\Big]_{-2}^{5} \)

    \( \displaystyle = \frac{49}{2} - 0 = \frac{49}{2} \)

Adding the two parts:

    \( \displaystyle \frac{9}{2} + \frac{49}{2} = 29 \)