Integration of Powers and Products of Sine and Cosine
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Lesson Summary:
In this lesson, we learn how to integrate powers and products of sine and cosine. We explore the different methods of integration for odd and even powers of sine and cosine, including using substitution, power reduction formulas, and integration by parts. We also learn how to handle products of factors with different periods using trigonometric product to sum formulas. By the end of the lesson, we have a comprehensive understanding of how to integrate various combinations of powers and products of sine and cosine.
Lesson Description:
Integral of (cos x)^m (sin x)^n dx. Also, integral of cos(ax) sin(bx) dx, etc.
Additional Resources:
- Table of Integrals - Table of integrals of a bunch of different functions.
- Special Functions - Table of some different special functions.
Questions answered by this video:
- What is the integral of (sin x)^m (cos x)^n?
- How do you find the integral of (cos x)^2 (sin x)^3?
- What is the integral of (cos x)^3 / sin x?
- What are the power reduction formulas?
- What is the integral of (sin x)^2?
- What are reduction formulas for integrals of powers of sin and cos?
- How do you find integrals of products of sin and cos factors with different periods?
- What are product-to-sum formulas?
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Staff Review
This video explains how to integrate power of sin and cos multiplied together, such as (cos x)^3. These techniques use u-substitution and using the fact that (sin x)^2 + (cos x)^2 = 1. In some problems, it is necessary to use double angle formulas to simplify and integrate. Some good examples are done in this video and many different techniques are explained. Finally, the entire lesson is summed up at the end of the video. -
honu07
I found this video extremely helpful. My math teacher did not explain integrating sine and cosine to powers very well at all! This video taught me everything I needed to know in a clear, concise manner with detailed examples at a slow enough pace that I was still able to follow and understand. -
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