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Integration using trigonometric identities

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Calculus 12
Limits and Continuity
29. Find antiderivatives of polynomials e^kx and selected trigonometric functions of kx
29.1 Integration using trigonometric identities

Integrating Trigonometric Functions Using Identities

Master the technique of rewriting trig expressions with identities so you can turn tricky powers of sine, cosine, and tangent into integrals you already know how to solve.

What You'll Learn

Recognize when a trig integral needs an identity rewrite before any basic rule applies
Apply the Pythagorean identity to split an odd power of sine or cosine for u-substitution
Use the power-reducing formulas to integrate even powers like sin squared x and cos squared x
Rewrite tangent squared or cotangent squared using the reciprocal identities before integrating
Combine double-angle identities with substitution to handle products of sine and cosine
Distinguish this identity-rewriting approach from trigonometric substitution used for algebraic integrands

What You'll Practice

1

Integrating products of sine and cosine with odd exponents using case 1 and 2

2

Applying half-angle identities to integrate even-powered trigonometric functions

3

Stripping factors and pairing with dx for effective u-substitution

4

Converting trigonometric expressions using Pythagorean identities before integrating

Why This Matters

Mastering trigonometric integration is essential for calculus and beyond, appearing frequently in physics, engineering, and advanced mathematics. These techniques form the foundation for solving complex problems involving periodic functions, wave equations, and Fourier analysis.

Before You Start — Make Sure You Can:

Other word problems relating angles in trigonometryDouble-angle identitiesPythagorean identitiesQuotient identities and reciprocal identitiesSum and difference identitiesCofunction identities

This Unit Includes

3 Video lessons
Practice exercises
Learning resources

Skills

U-Substitution
Trigonometric Identities
Half-Angle Identities
Pythagorean Identity
Integration
Antiderivatives
Double-Angle Formula
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