The value of ∫_-5^5 |x+3| dx is

The correct answer is Option (3) → 34

The expression inside the modulus changes sign at x = −3, so split the integral there:

    \(|x + 3| = -(x + 3)\) for \(-5 \le x \le -3\)
    \(|x + 3| = x + 3\) for \(-3 \le x \le 5\)

    \( \displaystyle \int_{-5}^{-3} -(x + 3)\,dx = \left[-\frac{(x + 3)^2}{2}\right]_{-5}^{-3} = 0 - \left(-\frac{4}{2}\right) = 2 \)

    \( \displaystyle \int_{-3}^{5} (x + 3)\,dx = \left[\frac{(x + 3)^2}{2}\right]_{-3}^{5} = \frac{64}{2} - 0 = 32 \)

    \( \displaystyle \int_{-5}^{5} |x + 3|\,dx = 2 + 32 = 34 \)

Shortcut: The graph of \(y = |x + 3|\) is a V with its vertex at (−3, 0). Over [−5, 5] the area under it is two triangles: one of base 2 and height 2, and one of base 8 and height 8.

    \( \displaystyle \frac{1}{2}(2)(2) + \frac{1}{2}(8)(8) = 2 + 32 = 34 \)