Questions: Suppose the integral from 3 to 5 of f(x) dx equals 5, the integral from 5 to 7 of f(x) dx equals -4, and the integral from 5 to 7 of g(x) dx equals 1. Evaluate the integrals in parts a - . a. The integral from 3 to 5 of 4 f(x) dx equals □ (Simplify your answer.)

Solution Steps

We are given the following integrals: \[ \int_{3}^{5} f(x) \, dx = 5, \quad \int_{5}^{7} f(x) \, dx = -4, \quad \int_{5}^{7} g(x) \, dx = 1. \] We are asked to evaluate the integral: \[ \int_{3}^{5} 4 f(x) \, dx. \]

The constant multiple rule states that: \[ \int_{a}^{b} k f(x) \, dx = k \int_{a}^{b} f(x) \, dx, \] where \( k \) is a constant. In this case, \( k = 4 \).

We know that: \[ \int_{3}^{5} f(x) \, dx = 5. \] Therefore: \[ \int_{3}^{5} 4 f(x) \, dx = 4 \cdot \int_{3}^{5} f(x) \, dx = 4 \cdot 5 = 20. \]

Final Answer