Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

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:

• 8.2.1 Evaluate integrals of the form int sin^mx cos^nx dx when (1) m is odd, (2) n is odd, or (3) m and n are even.
• 8.2.2 Evaluate integrals of the form int tan^mx sec^nx dx when (1) m is odd, (2) n is even, (3) m even with n odd, (4) m=0, or (5) n=0.
• 8.2.3 Use product-to-sum trigonometric identities to evaluate integrals of the form int sin(ax)cos(bx)dx.

Terminology & Tips

 Rules for Evaluating Trigonometric Integrals Evaluate int sin^mx cos^nx dx. Odd sine. Separate one sinx out for du, convert rest to cosines using Pythagorean id, use sub u=cosx. Odd cosine. Separate one cosx out for du, convert rest to sines using Pythagorean id, use sub u=sinx. 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. Odd tangent. Separate secx * tanx out for du, convert rest to secants using Pythagorean id, use sub u=secx. Even secant. Separate out sec^2x for du, convert rest to tangents using Pythagorean id, use sub u=tanx. 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

Supplemental Resources (optional)

rev. 2021-01-30