Question

question 7 (1 point)
suppose $\int_{-2}^{7}(3 f(x)) d x=12$ and $\int_{5}^{7} f(x) d x=2$. compute $\int_{-2}^{5} f(x) d x$.
4
8
10
0
2
view hint for question 7

Explanation:

Step1: Use the property of definite integral \(\int_{a}^{b}kf(x)dx = k\int_{a}^{b}f(x)dx\)

Given \(\int_{-2}^{7}(3f(x))dx = 12\), then \(3\int_{-2}^{7}f(x)dx=12\), so \(\int_{-2}^{7}f(x)dx = 4\)

Step2: Use the property of definite integral \(\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx\)

We know that \(\int_{-2}^{7}f(x)dx=\int_{-2}^{5}f(x)dx+\int_{5}^{7}f(x)dx\)
Since \(\int_{-2}^{7}f(x)dx = 4\) and \(\int_{5}^{7}f(x)dx = 2\)
Substitute the values into the formula: \(4=\int_{-2}^{5}f(x)dx + 2\)

Step3: Solve for \(\int_{-2}^{5}f(x)dx\)

\(\int_{-2}^{5}f(x)dx=4 - 2\)

Answer:

\(2\)