Integration of Powers and Products of Secant and Tangent, Cosecant and Cotangent

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Taught by Houston
  • Currently 4.0/5 Stars.
6652 views | 2 ratings
Meets NCTM Standards:
Lesson Summary:

This lesson covers the integration of powers and products of secant and tangent, as well as cosecant and cotangent. The lesson covers various techniques, including substitution and integration by parts, to find the integrals of functions such as secant cubed x dx and tangent to the fifth x dx. The reduction formulas for odd powers of secant are derived and used in examples, and special tricks for finding the integrals of even powers of secant are also explained. The lesson provides a comprehensive overview of techniques for integrating these types of functions.

Lesson Description:

Integral of (sec x)^m (tan x)^n dx and integral of (csc x)^m (cot x)^n dx.

Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at

Additional Resources:
Questions answered by this video:
  • What is the integral of (sec x)^m (tan x)^n?
  • How do you find the integral of (sec x)^2?
  • What is the integral of tan x?
  • What is the integral of sec x?
  • How do you find the integral of (tan x)^2?
  • How do you find the integral of (tan x)^2 (sec x)^4?
  • How do you find the integral of (tan x)^3 (sec x)^3?
  • What is the reduction formula for powers of tangent?
  • What is power reduction via integration by parts for odd powers of secant?
  • Staff Review

    • Currently 4.0/5 Stars.
    This video is very similar to the integration of powers of sine and cosine. Some very useful and important tricks and results are obtained. Some of the derivations are very complicated and done in-depth, but the results and examples are incredibly useful and interesting. Again, near the end of the video, the lesson is summed up. Finally, cosecant and cotangent are discussed briefly. Although it can be confusing at times, this video is very important and certain results are vital to remember for integration.