What method is used to simplify integrals through substitution?

The method used to simplify integrals through substitution is known as integration by substitution. This technique is particularly effective when dealing with integrals that contain composite functions.

The process involves identifying a substitution that simplifies the integral into a more manageable form. By choosing a function within the integral to represent a new variable, you effectively transform the integral into a simpler form that is easier to evaluate. After performing the integration with respect to the new variable, you can substitute back to the original variable at the end of the process.

For example, if you have an integral of the form ∫f(g(x))g'(x)dx, you can let u = g(x). The differential du then corresponds to g'(x)dx, allowing you to rewrite the integral as ∫f(u)du. This makes solving the integral straightforward.

The other methods mentioned do not align with the principles of substitution. Integration by parts is used for integrating the product of two functions, integration by constants is not a recognized method, and integration by distribution does not correctly pertain to the techniques used in calculus for integration. Therefore, the choice of integration by substitution is not only valid but is an essential technique for simplifying certain integrals.