Question

question 11
5 pts
suppose that $\int_{-1}^{4} f(x) d x=5, \int_{4}^{7} f(x) d x=-6$, and $\int_{-1}^{7} g(x) d x=2$. compute: $\int_{-1}^{7}(5 f(x)-g(x)) d x$.
-7
-3
1
no correct answer choice is given.
9

Explanation:

Step1: Use integral properties

Use \(\int_{a}^{b}(cf(x)-dg(x))dx = c\int_{a}^{b}f(x)dx - d\int_{a}^{b}g(x)dx\) and \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\). So \(\int_{-1}^{7}f(x)dx=\int_{-1}^{4}f(x)dx+\int_{4}^{7}f(x)dx\).
Substitute \(\int_{-1}^{4}f(x)dx = 5\) and \(\int_{4}^{7}f(x)dx=-6\), then \(\int_{-1}^{7}f(x)dx=5+( - 6)=-1\).

Step2: Compute the target integral

\(\int_{-1}^{7}(5f(x)-g(x))dx = 5\int_{-1}^{7}f(x)dx-\int_{-1}^{7}g(x)dx\).
Substitute \(\int_{-1}^{7}f(x)dx=-1\) and \(\int_{-1}^{7}g(x)dx = 2\).
We get \(5\times(-1)-2=-5 - 2=-7\).

Answer:

\(-7\)