If $\int_{3}^{-1} f(x) d x=-2$ and $\int_{5}^{3} f(x) d x=7$, then find : $\int_{5}^{-1} f(x) d x$

Step 1 ：Given that \(\int_{3}^{-1} f(x) d x=-2\) and \(\int_{5}^{3} f(x) d x=7\)

Step 2 ：Using the property of definite integrals, we know that \(\int_{3}^{-1} f(x) d x\) is equal to \(-\int_{-1}^{3} f(x) d x\)

Step 3 ：Also, the integral of a function from a to c can be found by adding the integral from a to b and the integral from b to c. So, \(\int_{5}^{-1} f(x) d x\) is equal to \(\int_{5}^{3} f(x) d x + \int_{3}^{-1} f(x) d x\)

Step 4 ：Substituting the given values, we get \(\int_{5}^{-1} f(x) d x = 7 - 2 = 5\)

Step 5 ：Final Answer: The value of \(\int_{5}^{-1} f(x) d x\) is \(\boxed{5}\)