Course: Calculus II
Topic: Integration Techniques
Subtopic: Trigonometric Substitutions


In this lesson we learn to evaluate integrals of certain rational expressions, by transforming them via substitutions obtained from trigonometric triangles, into a trigonometric integral and then applying those processes to finish off the evaluation. This technique is called the method of trigonometric substitutions. The substitutions are based relationships obtained by three trigonometric triangles which you should memorize.


By the end of this topic you should know and be prepared to be tested on:

Terminology & Tips

Terms you should be able to define: trigonometric triangle, trigonometric substitution

Sketches of the three trig triangles used in the trigonometric substitution method. If triangle has radical u-squared plus a-squared use u=a times tangent theta. If radical a-squared minus u-squared use u=a times sine theta. If radical u-squared minus a-squared use u=a secant theta.

Mini-Lectures and Examples

STUDY: Integration Techniques - Trigonometric Substitutions

Supplemental Resources (optional)

Video: Trigonometric Substitutions, Selwyn Hollis's Video Calculus

Lesson: Trigonometric Substitution - Another Change of Variable, Dale Hoffman's Contemporary Calculus

rev. 2021-02-06