# UTPA STEM/CBI Courses/Calculus/Trigonometric Integrals and Substitutions

Course Title: Calculus II

Lecture Topic: Trigonometric Integrals and Substitutions

Instructor: Baofeng Feng/Wei Yin

Institution: UTRGV/STC

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Solve for several types of integrals involving trigonometric functions;
• Solve integrals that can be transformed into trigonometric integrals by substitutions.
• Sub-Objectives- The objectives will require that students be able to:
• Become familiar with relations and identities of trigonometric functions;
• Learn the derivatives and anti-derivatives of trigonometric functions;
• Understand the definition of trigonometric and inverse trigonometric functions based on a triangle.
• Difficulties- Students may have difficulty:
• Remembering many of the formulas and identities of trigonometric functions.
• Differentiating between derivatives and anti-derivatives of trigonometric functions.
• Recognizing different types of trigonometric integrals that are required to successfully complete this section;
• Determining when to use substitution or integration by parts when trying to complete an integration problem.
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• Find the area of the ellipse

Model of Knowledge

• Concept Map
• Definition of trigonometric and inverse functions.
• Definition of inverse-trigonometric functions and their solutions.
• Content Priorities
• Enduring Understanding
• Solving integrals of trigonometric functions by substitutions, integrations by parts and trigonometric substitutions.
• Determine which trigonometric substitution is appropriate for different integrals.
• Write the solution in a form that uses the original variables.
• Important to Do and Know
• The trigonometric relations and identities such as the half-angle formulas, the double-angle formulas, and the three fundamental identities.
• Recognize the pattern and choose the right substitution formulas.
• Worth Being Familiar with
• Different types of trigonometric integrals.
• Different patterns of integrals which can be converted into trigonometric integrals.

Assessment of Learning

• Formative Assessment
• In Class (groups)
• Distribute a working sheet for them to recognize the types of trigonometric integrals without actually solving them
• Distribute a working sheet for them to recognize the types of integrals for trigonometric substitutions without actually solving them
• Homework (individual)
• Using Webwork to assign online homework problems.
• Summative Assessment
• In class quiz right after the section is complete.
• In class test after the completion of this chapter.
• Final exam to check their final mastery of this section.

## Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Solve typical types of trigonometric integrals and integrals which need to be converted into trigonometric integrals.
• Remember and recognize the types of trigonometric integrals and integrals for trigonometric substitutions.

The objectives will require that students be able to:

• Solve several types of trigonometric integrals and integrals with trigonometric substitutions
• Confidently use trigonometric and inverse trigonometric definitions, relations and identities to deal with trigonometric integrals and integrals with trigonometric substitutions.

THE CHALLENGE

Problems will be given which can be solved using regular or trigonometric substitution. Through several iterations, it should become clear that either method will result in the same integral given proper application.

GENERATE IDEAS

Once the problem is presented, the students will be given a few minutes to internalize the problem. They can brainstorm possible paths to a solution.

MULTIPLE PERSPECTIVES

The students will work in small groups for 10 min and come up with ideas.

RESEARCH & REVISE

The instructor will solve the problem for the students using two different methods. This will encourage those with positive ideas and allow those that were not on the right track to see where mistakes can be corrected.

The instructor will use internet resources to assign homework of different problem types.

GO PUBLIC

One student (volunteer) will solve the challenge on the white board.

## Pre-Lesson Quiz

Various material can be tested here, depending on time constraints.