Trigonometric Substitutions

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Presentation transcript:

Trigonometric Substitutions Section 8.4b

Trigonometric Substitutions How do we evaluate this integral? These trigonometric substitutions allow us to replace binomials of the form by single squared terms, and thereby transform a number of integrals (like the one above) into ones we can evaluate directly or find in a table of integrals.

Trigonometric Substitutions The most common substitutions are based on the following reference triangles: With

Trigonometric Substitutions The most common substitutions are based on the following reference triangles: With

Trigonometric Substitutions The most common substitutions are based on the following reference triangles: With

Trigonometric Substitutions 1. replaces with 2. replaces with 3. replaces with Also, we want any substitution to be reversible so we can change back to the original variable afterward. For example: requires Essentially, keep positives with any absolute values… with

Trigonometric Substitutions Set Evaluate

Trigonometric Substitutions Set Evaluate

Trigonometric Substitutions Evaluate Appendix A7, Formula 88 (p.631)

Trigonometric Substitutions Evaluate

Trigonometric Substitutions Evaluate