Why does trigonometric substitution work?

Abstract

ABSTRACTModern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique of integration. In this article, we present an accessible proof that establishes the validity of integration by trigonometric substitution. The proof offers calculus instructors a simple argument that can be used to show their students that trigonometric substitution is a valid technique of integration.