∫_2^5 |x - 3| dx equals

The correct answer is Option (2) → \(\displaystyle\frac{5}{2}\)

The expression inside the modulus changes sign at \(x = 3\):

    \(|x - 3| = 3 - x\) for \(2 \le x \le 3\)
    \(|x - 3| = x - 3\) for \(3 \le x \le 5\)

So the integral is split at x = 3:

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

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

    \( \displaystyle = \left(\frac{9}{2} - 4\right) + \left(-\frac{5}{2} + \frac{9}{2}\right) \)

    \( \displaystyle = \frac{1}{2} + 2 = \frac{5}{2} \)

Shortcut: The graph of \(y = |x - 3|\) on [2, 5] forms two triangles above the x-axis: base 1 and height 1, and base 2 and height 2. Their areas are \(\frac{1}{2}\) and 2, so the integral is \(\frac{5}{2}\).

Why the other options are incorrect

• Option (1) \(\displaystyle\frac{3}{2}\) — Incorrect. This is \(\int_{2}^{5}(x - 3)\,dx\), the value without the modulus, where the part on [2, 3] counts as negative.

• Option (3) \(\displaystyle\frac{1}{2}\) — Incorrect. This is only the part from 2 to 3.

• Option (4) 5 — Incorrect. The two parts add up to \(\frac{5}{2}\), not 5.