Why does trigonometric substitution work?

ABSTRACT

Modern 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.

KEYWORDS:

• Calculus instruction
• technique of integration
• inverse substitution

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Let J be the domain of f. It follows that the inverse function g⁻¹: J → I exists and is differentiable on J.

2. Note that G has domain I.

3. The derivative at an endpoint is the appropriate one-sided limit of the difference quotient.