1 multiple choice 1 point. if ∫₀¹f(x)dx = 2 and ∫₀⁴f(x)dx = -3, then ∫₁⁴(3f(x)+2)dx = (a) -13 (b) -9 (c) -7…

Answer:

A. -13

Explanation:

Step1: Use integral property

We know that (\int_{0}^{4}f(x)dx=\int_{0}^{1}f(x)dx+\int_{1}^{4}f(x)dx). Given (\int_{0}^{1}f(x)dx = 2) and (\int_{0}^{4}f(x)dx=-3), we can find (\int_{1}^{4}f(x)dx) as (\int_{1}^{4}f(x)dx=\int_{0}^{4}f(x)dx-\int_{0}^{1}f(x)dx=-3 - 2=-5).

Step2: Expand the integral

(\int_{1}^{4}(3f(x)+2)dx=3\int_{1}^{4}f(x)dx+\int_{1}^{4}2dx).

Step3: Calculate each integral

First, (3\int_{1}^{4}f(x)dx = 3\times(-5)=-15). Second, (\int_{1}^{4}2dx=2x\big|_{1}^{4}=2\times(4 - 1)=6).

Step4: Combine results

(3\int_{1}^{4}f(x)dx+\int_{1}^{4}2dx=-15 + 6=-9).