Essential Concepts
- Integrals of trigonometric functions can be evaluated by the use of various strategies. These strategies include:
- Applying trigonometric identities to rewrite the integral so that it may be evaluated by u-substitution
- Using integration by parts
- Applying trigonometric identities to rewrite products of sines and cosines with different arguments as the sum of individual sine and cosine functions
- Applying reduction formulas
Key Equations
To integrate products involving [latex]\sin\left(ax\right)[/latex], [latex]\sin\left(bx\right)[/latex], [latex]\cos\left(ax\right)[/latex], and [latex]\cos\left(bx\right)[/latex], use the substitutions.
- Sine Products
[latex]\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)[/latex] - Sine and Cosine Products
[latex]\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)[/latex] - Cosine Products
[latex]\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)[/latex] - Power Reduction Formula
[latex]{\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[/latex] - Power Reduction Formula
[latex]{\displaystyle\int}{\tan}^{n}xdx=\frac{1}{n - 1}{\tan}^{n - 1}x-{\displaystyle\int}{\tan}^{n - 2}xdx[/latex]
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
