Lesson 27: Integrals of Transcendental Functions

Preview

Here's the on-ramp for this lesson.


What Will We Learn?

  • This lesson wraps up our calculus adventure by discussing integration within the context of transcendental functions. For this family of functions, we’ll revisit the Fundamental Theorem of Calculus, u-substitution, and some applications of integration.


Why Do We Need to Learn This?

  • After this lesson, we will have covered and learned all the core content in a typical calculus course. The integration of transcendental functions, in particular, fills in some important gaps in our study of integration. One example: the indefinite integral of 1/x. This is a case not covered by the integral version of the Power Rule, and ends up yielding a logarithm. More broadly, the integration of transcendental functions has broad real-applications—as we’ll learn in this lesson—and also forms the foundation for advanced mathematics and engineering applications (e.g., Fourier Analysis).

Review

Learn

The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:

  • I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.

Lesson Notes

Lesson 27.pdf

Video 1 (Examples 5.16-5.19)

Video 2 (Examples 5.25-5.27)

Reflect

If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:

  • Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.

  • Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.

If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:

  • Please summarize the main mathematical takeaways from the lesson notes.

  • What was the most interesting part of what you learned, and why?

  • What, if anything, do you still find confusing?

Practice

Work through the practice problems suggested below to see how much of this lesson you've understood.

Lesson 27 PP.pdf