Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Antiderivatives
Subtopic: Integration by Substitution & Definite Integrals

Overview

We have used the u-substitution method to evaluate indefinite integrals, but here we use it to evaluate definite integrals. Note that with indefinite integrals at the end of the process we convert back to in terms of x, but with definite integrals we convert everything to in terms of u, including the limits of integration, and never return to x.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 4.7.1 Recognize that u-substitution is required and find an appropriate u-substitution
• 4.7.2 Using u-substitution with DEFINITE integrals: Rewrite the integral from in terms of x to an equivalent integral in terms of u and du, remember to convert the limits of integration too, then evaluate the new integral with respect to u, evaluate using the u-limits to get the final answer
• 4.7.3 Use algebraic techniques (e.g. complete the square, use trig identities) to transform the integrand to an expression where the u-substitution method will work

Terminology

Terms you should be able to define: convert limits of integration to "in terms of u"

Mini-Lectures and Examples

Supplemental Resources (optional)

Lesson: First Applications of Definite Integrals, Dale Hoffman's Contemporary Calculus includes a variety of applications for motivation for the transition from Calc I to Calc II.

rev. 2020-11-27