Which case of integration by substitution involves substituting a function directly into the integrand?

The first case of integration by substitution typically involves directly substituting a function into the integrand. This process is utilized when you identify an inner function within the integrand that can be substituted with a single variable to simplify the integration. In this case, you let ( u = f(x) ), and for the differential ( du ), you derive it from ( dx ), allowing you to express the entire integral in terms of ( u ).

For example, if you have an integral such as ( \int f(g(x)) g'(x) , dx ), you set ( u = g(x) ) and substitute this into the integrand. The advantage of this approach is that it transforms the integral into a simpler form, often making it easier to evaluate.

This method of direct substitution is particularly useful when the integrand is composed of a function and its derivative, as is commonly seen in many integral problems. Other cases, while also valid methods of integration, do not emphasize this direct form of substitution into the integrand as the primary technique.