Summary of Trigonometric Integrals

Essential Concepts

Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies include:
1. Applying trigonometric identities to rewrite the integral so that it may be evaluated by u-substitution
2. Using integration by parts
3. Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions
4. Applying reduction formulas

Key Equations

To integrate products involving $\sin\left(ax\right)$ , $\sin\left(bx\right)$ , $\cos\left(ax\right)$ , and $\cos\left(bx\right)$ , use the substitutions.

Sine Products
$\sin\left(ax\right)\sin\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)-\frac{1}{2}\cos\left(\left(a+b\right)x\right)$
Sine and Cosine Products
$\sin\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\sin\left(\left(a-b\right)x\right)+\frac{1}{2}\sin\left(\left(a+b\right)x\right)$
Cosine Products
$\cos\left(ax\right)\cos\left(bx\right)=\frac{1}{2}\cos\left(\left(a-b\right)x\right)+\frac{1}{2}\cos\left(\left(a+b\right)x\right)$
Power Reduction Formula
${\displaystyle\int}{\text{sec}}^{n}xdx=\frac{1}{n - 1}{\text{sec}}^{n - 1}x+\frac{n - 2}{n - 1}{\displaystyle\int}{\text{sec}}^{n - 2}xdx$
Power Reduction Formula
${\displaystyle\int}{\tan}^{n}xdx=\frac{1}{n - 1}{\tan}^{n - 1}x-{\displaystyle\int}{\tan}^{n - 2}xdx$

Glossary

power reduction formula: a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

trigonometric integral: an integral involving powers and products of trigonometric functions

Module 3: Techniques of Integration

Summary of Trigonometric Integrals

Essential Concepts

Key Equations

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