If $f(x)=8 x-7$ for $x \leq 3$ and $f(x)=4 g(x)$ for $x> 3$ where $\int_{3}^{15} g(x) d x=4$ then $\int_{0}^{15} f(x) d x=$

Step 1 ：Split the integral of \(f(x)\) from 0 to 15 into two integrals: from 0 to 3 and from 3 to 15.

Step 2 ：For the first integral, use the function \(f(x) = 8x - 7\).

Step 3 ：For the second integral, use the function \(f(x) = 4g(x)\) and the given that the integral of \(g(x)\) from 3 to 15 is 4.

Step 4 ：Calculate these two integrals separately and then add them together to get the final answer.

Step 5 ：The integral of \(f(x)\) from 0 to 3 is 15.

Step 6 ：The integral of \(f(x)\) from 3 to 15 is 192.

Step 7 ：Add these two integrals together to get the final answer: \(15 + 192 = 207\).

Step 8 ：Final Answer: The integral of \(f(x)\) from 0 to 15 is \(\boxed{207}\).