For example, in calculus, the intermediate function $u = x^2$ is often used to simplify integrals of the form $\int x^n dx$. By making the substitution $u = x^2$, the integral becomes $\int u^{n/2} du$, which is a simpler integral to evaluate.
Intermediate functions can also be used to solve differential equations. For example, the integrating factor for a first-order linear differential equation is an intermediate function that is used to transform the equation into a separable equation, which can then be solved by separation of variables.
Intermediate functions are a powerful tool that can be used to solve a wide variety of problems in mathematics and science. They provide a way to break down complex problems into simpler steps, and to make difficult integrals and differential equations more tractable.