Question

$\int_{1}^{5} \left(|x - 3| + |1 - x|\right) \, dx =$

• A. $12$
• B. $\frac{5}{6}$
• C. $21$
• D. $10$

Answer 1

The Correct Option is A

Split the integral based on the behavior of the absolute values: 1. For $x \in [1, 3]$, $|x - 3| = 3 - x$ and $|1 - x| = x - 1$. 2. For $x \in [3, 5]$, $|x - 3| = x - 3$ and $|1 - x| = x - 1$. The integral becomes: \[ \int_{1}^{5} \left(|x - 3| + |1 - x|\right) dx = \int_{1}^{3} \left((3 - x) + (x - 1)\right) dx + \int_{3}^{5} \left((x - 3) + (x - 1)\right) dx. \] Simplify each part: 1. For $x \in [1, 3]$: \[ \int_{1}^{3} \left((3 - x) + (x - 1)\right) dx = \int_{1}^{3} (2) dx = 2(3 - 1) = 4. \] 2. For $x \in [3, 5]$: \[ \int_{3}^{5} \left((x - 3) + (x - 1)\right) dx = \int_{3}^{5} (2x - 4) dx = \left[x^2 - 4x\right]_{3}^{5}. \] Evaluate: \[ \left[x^2 - 4x\right]_{3}^{5} = (25 - 20) - (9 - 12) = 5 + 3 = 8. \] Add the results: \[ 4 + 8 = 12. \]