What case focuses on the method of integrating a function along with its derivative?

The case that focuses on the method of integrating a function along with its derivative is indeed the third case of Integration by Substitution. This technique is particularly useful when you encounter an integral that involves a function and its derivative simultaneously, allowing you to simplify the integral effectively.

In this case, the key is to identify a substitution that relates the original function to its derivative. For instance, if you have an integral of the form ∫f(g(x))g'(x)dx, you can let u = g(x), thereby transforming the integral into a simpler form ∫f(u)du. This change of variables leads to a direct computation of the integral in terms of the new variable.

In general, the principle behind this method is that it allows you to transform complex integrals involving derivatives into simpler integrals that can be solved more easily. This alignment of a function with its derivative is what characterizes this particular case in Integration by Substitution.

Other options may reference different techniques or special cases that do not specifically emphasize the relationship between a function and its derivative in the same way that this case does.