Integrating with Special Substitutions
In this lesson, Professor Perez teaches about substitutions for rational functions of sine and cosine, using Carl Weierstrass's method. Using double angle formulas, reference triangles, and inverse trig functions, he shows how to replace sine x, cosine x, and dx with expressions in terms of u, where u equals tangent of x over 2. With this substitution, he demonstrates how to simplify an integral and solve for the denominator of a quadratic expression, potentially leading to either trig substitution or partial fraction decomposition.
Rational functions with sine and cosines: Karl Weierstrass method.
Created by and copyright of Larry Perez. Funded by the state of California through Saddleback College. More information on videos, resources, and lessons at Algebra2Go.
- What are special substitutions in integration?
- How do you take the integral of rational functions with sine and cosine?
- What is the Karl Weierstrass method?
- What is trig substitution?
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Staff Review
The Karl Weierstrass method is explained, the procedure is outlined, and special substitutions are used in order to integrate some rational functions with sine and cosine in them. The double angle formulas are used to derive the substitutions for trig formulas. The substitution for tangent is also shown.
