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.)

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: \(\boxed{42.05}\)