In which case of integration by substitution do you use a shortcut method for integrating a function and its derivative?

The shortcut method for integrating a function and its derivative is applicable in the context of integration by substitution, specifically in the second case. This technique is particularly effective when you have an integral of the form ∫f(g(x))g'(x)dx. Here, f(g(x)) is a function and g'(x) is its derivative.

In this scenario, you recognize that you can substitute g(x) as a new variable, usually denoted as u. By doing so, the integral simplifies significantly: the derivative g'(x)dx corresponds directly to the differential du, allowing you to rewrite the integral in terms of u. The next step is to integrate f(u) with respect to u, which often leads to a solution that is straightforward since f and its integral are typically easier to compute than the original function.

This method is particularly valuable because it leverages the relationship between a function and its derivative, streamlining the integration process. It allows the substitution of complex expressions into a more manageable format, illustrating a common technique that enhances efficiency in solving integrals involving derivatives directly. The effectiveness of this shortcut exemplifies why it is included in the second case of integration by substitution.

In contrast, the other options may involve more complex relationships or