Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the importance of recognizing and fitting integrals to the basic integration rules through algebraic manipulation and substitution techniques. Students learn to identify the correct rule—whether it is the power rule, Log Rule, Arctangent Rule, or other trigonometric-based rules—by analyzing the structure of the integrand. Furthermore, the section highlights common techniques such as dividing improper rational functions, expanding numerators, and employing trigonometric identities to simplify complex integrals before applying standard formulas.

Learning Objectives

1

Identify which basic integration rule is applicable to a given integrand by analyzing its structure.

2

Apply algebraic manipulation techniques (such as long division, completing the square, and adding/subtracting terms) to rewrite integrals in a recognizably standard form.

3

Use substitution, including trigonometric substitution and u?substitution, to fit an integrand to a basic integration rule.

4

Integrate functions by combining multiple basic rules (e.g., the power rule, Log Rule, Arctangent Rule, and Arcsine Rule) when necessary.

5

Recognize and avoid common mistakes in separating numerators and denominators or misapplying integration rules.

Key Concepts

CONCEPT

DEFINITION

Basic Integration Rules

A set of standard formulas used to find antiderivatives of functions, including rules like the power rule, Log Rule, Arctangent Rule, and Arcsine Rule.

Arctangent Rule

An integration formula used for integrals that have the form ∫du/(a^2 + u^2) leading to an antiderivative of (1/a) arctan(u/a) + C.

Log Rule

A rule used when the integral includes a quotient where the derivative of the denominator’s inner function appears in the numerator, resulting in a natural logarithm in the antiderivative.

Substitution

A method for simplifying integrals by making a change of variable (u = g(x)) so that the integrand fits a basic integration rule.

Trigonometric Identities

Algebraic formulas involving trigonometric functions that allow transformation of an integrand into a form that matches a basic integration rule.

Improper Rational Functions

Rational functions where the degree of the numerator is greater than or equal to the degree of the denominator; these often require algebraic division before integration.

Example Problems

Example 1

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative. $\frac{d y}{d x}=\frac{x}{\sqrt{x^{2}+1}}$ $$ \begin{array}{ll}{\text { (a) } 2 \sqrt{x^{2}+1}+C} & {\text { (b) } \sqrt{x^{2}+1}+C} \\ {\text { (c) } \frac{1}{2} \sqrt{x^{2}+1}+C} & {\text { (d) } \ln \left(x^{2}+1\right)+C}\end{array} $$

Example 2

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative. $\frac{d y}{d x}=\frac{x}{x^{2}+1}$ $$ \begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+c} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+c} \\ {\text { (c) arctan } x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array} $$

Example 3

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative. $\frac{d y}{d x}=\frac{1}{x^{2}+1}$ $$ \begin{array}{ll}{\text { (a) } \ln \sqrt{x^{2}+1}+C} & {\text { (b) } \frac{2 x}{\left(x^{2}+1\right)^{2}}+C} \\ {\text { (c) } \arctan x+c} & {\text { (d) } \ln \left(x^{2}+1\right)+c}\end{array} $$

Example 4

Choosing an Antiderivative In Exercises $1-4,$ select the correct antiderivative. $\frac{d y}{d x}=x \cos \left(x^{2}+1\right)$ $$ \begin{array}{ll}{\text { (a) } 2 x \sin \left(x^{2}+1\right)+C} & {\text { (b) }-\frac{1}{2} \sin \left(x^{2}+1\right)+C} \\ {\text { (c) } \frac{1}{2} \sin \left(x^{2}+1\right)+C} & {\text { (d) }-2 x \sin \left(x^{2}+1\right)+C}\end{array} $$

Example 5

Choosing a Formula In Exercises $5-14$ , select the basic integration formula you can use to find the integral, and identify $u$ and $a$ when appropriate. $$ \int(5 x-3)^{4} d x $$
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Step-by-Step Explanations

QUESTION

How can you evaluate an integral resembling ∫(4x/(x^2-9)) dx when the denominator can be expressed as a sum/difference of squares?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the integrand may be manipulated to resemble the form required for the Arctangent Rule. Check if the substitution u = x (or a function of x) makes the numerator match the derivative of the denominator’s inner function.
Step 2: In cases where the numerator has extra factors, consider factoring out constants or using the Constant Multiple Rule.
Step 3: Rewrite the integrand so that it fits the standard form du/(a^2 + u^2) or a variant thereof.
Step 4: Apply the Arctangent Rule, which gives an antiderivative of (1/a) arctan(u/a) + C.
Final Answer: The resulting antiderivative is given in a form such as (4/3) arctan(x/3) + C (after appropriate algebraic manipulation).

Fitting an Integrand to the Arctangent Rule

QUESTION

How do you evaluate an integral that does not immediately appear to fit any basic rule but can be manipulated to use the Log Rule?

STEP-BY-STEP ANSWER:

Step 1: Examine the integrand to see if the numerator is related to the derivative of a function in the denominator.
Step 2: Add and subtract an appropriate term in the numerator to rewrite the integrand as a sum of two fractions.
Step 3: Separate the integrand into two simpler integrals, one of which fits the Log Rule.
Step 4: Integrate each part using the appropriate rule (Log Rule for the correct fraction and another rule, if necessary, for the remaining term).
Final Answer: Combine the integrated parts to obtain the final antiderivative in terms of natural logarithms and additional functions, plus an arbitrary constant C.

Using a Disguised Log Rule

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Common Mistakes

  • Mistakenly separating denominators when only the numerator is divisible or expandable.
  • Forgetting to adjust the differential (du) after making a substitution, leading to incorrect antiderivatives.
  • Incorrect algebraic manipulation such as failing to properly complete the square or conduct long division on improper fractions.
  • Misidentifying the applicable rule; for instance, attempting to apply the Arctangent Rule when the numerator does not match the derivative of the inner function.
  • Neglecting to include the constant of integration (C) in indefinite integrals.