3. (e) It is known that \( \int_{1}^{5} f(x) d x=5 \), and \( \int_{3}^{5} 2 f(x) d x=2 \), what does \( \int_{1}^{3}(f(x)-2) d x \) equal to ? A 1 B 3 C 4 D 0

Given: \[ \int_{1}^{5} f(x) \, dx = 5 \] and \[ \int_{3}^{5} 2f(x) \, dx = 2. \] **Step 1: Find \(\int_{3}^{5} f(x) \, dx\).** Since the integral is linear, we have: \[ \int_{3}^{5} 2f(x) \, dx = 2 \int_{3}^{5} f(x) \, dx. \] Therefore, we can solve for \(\int_{3}^{5} f(x) \, dx\): \[ 2 \int_{3}^{5} f(x) \, dx = 2 \quad \Rightarrow \quad \int_{3}^{5} f(x) \, dx = \frac{2}{2} = 1. \] **Step 2: Compute \(\int_{1}^{3} f(x) \, dx\).** Using the property of integrals over adjacent intervals: \[ \int_{1}^{5} f(x) \, dx = \int_{1}^{3} f(x) \, dx + \int_{3}^{5} f(x) \, dx, \] thus: \[ 5 = \int_{1}^{3} f(x) \, dx + 1 \quad \Rightarrow \quad \int_{1}^{3} f(x) \, dx = 5 - 1 = 4. \] **Step 3: Evaluate \(\int_{1}^{3} (f(x) - 2) \, dx\).** The integral can be separated as: \[ \int_{1}^{3} (f(x) - 2) \, dx = \int_{1}^{3} f(x) \, dx - \int_{1}^{3} 2 \, dx. \] We already found: \[ \int_{1}^{3} f(x) \, dx = 4. \] And the integral of the constant function \(2\) is: \[ \int_{1}^{3} 2 \, dx = 2(3 - 1) = 4. \] Thus: \[ \int_{1}^{3} (f(x) - 2) \, dx = 4 - 4 = 0. \] The correct answer is D \(0\).