How do you know to use the 100 as your c, as you are given an integral with 10 as its limit? How would you know to choose between the 10 and 100?

Because it is given in the question.

We are told

\(\displaystyle \int_{100}^{-5} f(x) dx = 4 \)

Which is the same thing as saying

\(\displaystyle - \int_{-5}^{100} f(x) dx = 4 \)

Going back to the question we want

\(\displaystyle \int_{-5}^{12} f(x) dx \)

Well...we know the value from -5 to 100, so lets use that. But to make this equal, we then need to go from 100 to 12! Therefore, we get

\(\displaystyle \int_{-5}^{12} f(x) dx = \int_{-5}^{100}f(x)dx + \int_{100}^{12} f(x) dx \)

We already know \(\displaystyle \int_{-5}^{100} f(x)dx = -4 \) but we don't quite know the integral of 100 to 12...so let's break that guy up in the same way.

\(\displaystyle \int_{100}^{12} f(x) dx = \int_{100}^{-10} f(x)dx + \int_{100}^{12} f(x) dx \)

Of course by reversing the limits we can find a numerical value.

To re-state...we choose our limits given the definitions in the original question