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Factoring Rational Expressions Rational Equations and Applications
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Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
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Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
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Calculus I
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15.Functions of Several Variables
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S = contains supplemental resources
Course: Calculus I
Topic: Antiderivatives
Subtopic: Integration by Substitution & Definite Integrals

Overview

Integration by substitution is a method that allows us to “undo” the chain rule. The process involves making a u-substitution that enables a somewhat complicated integral to be rewritten as one that is simpler and can be easily integrated. The key lies in making the correct substitution!

We have used 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:

• 4.6.1 Recognize that u-substitution is required and find an appropriate u-substitution
• 4.6.2 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
• 4.6.3 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

Define: convert limits of integration to "in terms of u"

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.