Knowing When to Avoid Trigonometry Substitution
Setting Up Partial Fractions When You Have Distinct Factors
When to Use Variable Substitution with Integrals

Using Identities to Express a Trigonometry Function as a Pair of Functions

You can express every product of powers of trig functions, no matter how weird, as the product of any pair of trig functions. The three most useful pairings are sine and cosine, tangent and secant, and cotangent and cosecant. The table shows you how to express all six trig functions as each of these pairings.


For example, look at the following function:


As it stands, you can’t do much to integrate this monster. But try expressing it in terms of each possible pairing of trig functions:


As it turns out, the most useful pairing for integration in this case is cot6 x csc2 x. No fraction is present — that is, both terms are raised to positive powers — and the cosecant term is raised to an even power.

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Integrating Powers of Cotangents and Cosecants
How to Integrate Odd Powers of Sines and Cosines
How to Integrate Compositions of Functions
Using Trigonometry Substitution to Integrate a Function
Setting Up Partial Fractions When You Have Distinct Linear Factors