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
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To solve the integral \( \int_{1}^{3}(f(x)-2) d x \), we can break it down using the properties of integrals. We can rewrite the integral as: \[ \int_{1}^{3} f(x) d x - \int_{1}^{3} 2 d x \] From the information given, we first need to find \( \int_{1}^{3} f(x) d x \). We know that: \[ \int_{1}^{5} f(x) d x = \int_{1}^{3} f(x) d x + \int_{3}^{5} f(x) d x = 5 \] We also know \( \int_{3}^{5} 2 f(x) d x = 2 \), which implies: \[ \int_{3}^{5} f(x) d x = 1 \] Now we can substitute that back into our first equation: \[ \int_{1}^{3} f(x) d x + 1 = 5 \] Thus, \[ \int_{1}^{3} f(x) d x = 5 - 1 = 4 \] Next, we calculate the second integral: \[ \int_{1}^{3} 2 d x = 2 \cdot (3 - 1) = 2 \cdot 2 = 4 \] Now we can combine the results: \[ \int_{1}^{3}(f(x)-2) d x = \int_{1}^{3} f(x) d x - \int_{1}^{3} 2 d x = 4 - 4 = 0 \] Therefore, the answer is: D 0