LESSON NOTES MENU
 
Course: Calculus II
Topic: Integration Techniques
Subtopic: Trigonometric Integrals

Overview

Our focus today is on evaluating trigonometric integrals, meaning integrals involving products and/or quotients of trigonometric functions such as `int sin^3x cos^2x dx`, `int sec^5x dx`, and `int sin^5x/cos^3x dx`. These trig integrals may require rewriting via identities, u-substitutions, and/or integration by parts. Each type of integral requires a particular plan of attack which we study in this lesson.

Objectives

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

Terminology & Tips

Rules for Evaluating Trigonometric Integrals

Evaluate `int sin^mx cos^nx dx`.

  1. Odd sine. Separate one `sinx` out for `du`, convert rest to cosines using Pythagorean id, use sub `u=cosx`.
  2. Odd cosine. Separate one `cosx` out for `du`, convert rest to sines using Pythagorean id, use sub `u=sinx`.
  3. Both even. Use half-angle id `sin^2x = 1/2 (1-cos2x)` or `cos^2x = 1/2 (1+cos2x)` to rewrite integral in form that can be evaluated.

Evaluate `int tan^mx sec^nx dx`.

  1. Odd tangent. Separate `secx * tanx` out for `du`, convert rest to secants using Pythagorean id, use sub `u=secx`.
  2. Even secant. Separate out `sec^2x` for `du`, convert rest to tangents using Pythagorean id, use sub `u=tanx`.
  3. Even tangent with odd secant or no tangents or no secants. Each integral requires a different approach. May require converting to all sines and cosines, integration by parts, or repeated use of rules 1 and 2.

Mini-Lectures and Examples

STUDY: Integration Techniques - Trigonometric Integrals

Supplemental Resources (optional)

Video: Integration of Powers and Products of Sine and Cosine, Selwyn Hollis's Video Calculus

Video: Integration of Powers and Products of Secant and Tangent and Cosecant and Cotangent, Selwyn Hollis's Video Calculus

Lesson: Integrals of Trig Functions, Dale Hoffman's Contemporary Calculus

rev. 2021-01-30