Use the properties of definite integrals to find $\int_{5} f(x) d x$ for the following function.
\[
f(x)=\left\{\begin{array}{ll}
6 x+7 & \text { if } x \leq 6 \\
-0.3 x+4 & \text { if } x> 6
\end{array}\right.
\]
\[
\int_{5}^{7} f(x) d x=\square
\]
(Simplify your answer.)
Final Answer: \(\boxed{42.05}\)
Step 1 :The integral of a piecewise function from a to b can be found by breaking the integral into two parts at the point where the function changes, which is at x=6 in this case. So, we need to find the integral of the function from 5 to 6 and from 6 to 7 separately and then add them together.
Step 2 :For the first part, we will integrate the function \(6x+7\) from 5 to 6. For the second part, we will integrate the function \(-0.3x+4\) from 6 to 7.
Step 3 :After finding the two integrals, we will add them together to get the final answer.
Step 4 :Let's calculate the two integrals.
Step 5 :The integral of the function \(6x+7\) from 5 to 6 is 40.
Step 6 :The integral of the function \(-0.3x+4\) from 6 to 7 is approximately 2.05.
Step 7 :The total integral is the sum of the two integrals, which is approximately 42.05.
Step 8 :The integral of the function from 5 to 7 is approximately 42.05.
Step 9 :Final Answer: \(\boxed{42.05}\)