Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

Group icon
127,035 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This textbook section focuses on developing and applying various techniques for evaluating integrals when antiderivatives are not immediately evident. Key methods include using basic integration formulas, substitution (often combined with algebraic manipulation such as completing the square), and integration by parts. The section also demonstrates the application of these methods through detailed examples and discusses the limitations of elementary antiderivatives, emphasizing the need for numerical methods and special techniques in certain cases.

Learning Objectives

1

Apply basic integration formulas and understand the process of finding antiderivatives.

2

Utilize the substitution method and algebraic techniques (including completing the square) to rewrite and evaluate integrals.

3

Develop strategies for handling integrals of specialized functions such as trigonometric and rational expressions.

4

Understand and implement integration by parts to simplify products of functions.

5

Recognize when numerical methods or special techniques are necessary for evaluating improper integrals and integrals without elementary antiderivatives.

Key Concepts

CONCEPT

DEFINITION

Definite Integral

An integral evaluated between specified limits, representing the net area under a curve.

Antiderivative

A function F such that F'(x) equals the given function f(x); used to evaluate definite integrals via the Fundamental Theorem of Calculus.

Substitution Method

A technique that involves changing the variable of integration to simplify an integral, often by matching it to a standard form.

Completing the Square

An algebraic method used to rewrite quadratic expressions in a perfect square form to facilitate integration.

Integration by Parts

A method derived from the product rule for differentiation, useful for integrating products of functions where one term is easily differentiated and the other is easily integrated.

Improper Integral

An integral with one or more infinite limits of integration or integrands that approach infinity, often requiring special techniques or numerical methods.

Long Division (of Rational Expressions)

A method used to rewrite improper rational functions as a polynomial plus a proper fraction, facilitating further integration.

Example Problems

Example 1

The integrals in Exercises $1-40$ are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int_{0}^{1} \frac{16 x}{8 x^{2}+2} d x $$

Example 2

The integrals in Exercises $1-40$ are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{x^{2}}{x^{2}+1} d x $$

Example 3

The integrals in Exercises $1-40$ are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int(\sec x-\tan x)^{2} d x $$

Example 4

The integrals in Exercises $1-40$ are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int_{\pi / 4}^{\pi / 3} \frac{d x}{\cos ^{2} x \tan x} $$

Example 5

The integrals in Exercises $1-40$ are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int \frac{1-x}{\sqrt{1-x^{2}}} d x $$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Evaluate ∫ from x=3 to x=5 (2x - 3)/(2x^2 - 3x + 1) dx.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the derivative of (x^2 - 3x + 1) is (2x - 3); set u = x^2 - 3x + 1 so that du = (2x - 3) dx.
Step 2: Change the limits of integration: when x = 3, u = 3^2 - 3(3) + 1 = 1; when x = 5, u = 5^2 - 3(5) + 1 = 11.
Step 3: Rewrite the integral in terms of u: ∫ from u=1 to u=11 du/(2u).
Step 4: Factor out the constant: (1/2) ∫ from u=1 to u=11 (du/u).
Step 5: Integrate to obtain (1/2)[ln |u|] evaluated from 1 to 11.
Final Answer: (1/2)[ln(11) - ln(1)] = (1/2) ln(11) since ln(1)=0.

Example 1 (Substitution Method)

QUESTION

Evaluate ∫ dx/(28x - x^2).

STEP-BY-STEP ANSWER:

Step 1: Rewrite the denominator: 28x - x^2 = -(x^2 - 28x).
Step 2: Complete the square for x^2 - 28x: x^2 - 28x = (x - 14)^2 - 196.
Step 3: Express the integrand as 1/(196 - (x - 14)^2) after adjusting the sign.
Step 4: Rewrite the integral to match the standard form ∫ dx/(a^2 - u^2) and apply an appropriate substitution, u = x - 14, with a = 14.
Step 5: Utilize the standard integration formula (from Table 8.1, Formula 18) to obtain the antiderivative.
Final Answer: The integrated result is sin⁻¹((x - 14)/14) + C (up to a constant factor as dictated by the substitution).

Example 2 (Completing the Square)

QUESTION

Evaluate ∫ ln x dx.

STEP-BY-STEP ANSWER:

Step 1: Choose u = ln x (since its derivative simplifies) and dv = dx.
Step 2: Differentiate u to get du = (1/x) dx, and integrate dv to obtain v = x.
Step 3: Apply the integration by parts formula: ∫ u dv = u*v - ∫ v du.
Step 4: Substitute the known values: = x ln x - ∫ x*(1/x) dx = x ln x - ∫ 1 dx.
Step 5: Integrate ∫ 1 dx to get x.
Final Answer: x ln x - x + C.

Example 3 (Integration by Parts)

Scroll left
Scroll right

Common Mistakes

  • Failing to adjust the limits of integration when performing substitution in definite integrals.
  • Incorrectly identifying the derivative factor required for the substitution, which can lead to mismatches in the integrand.
  • Errors in completing the square, such as miscalculating the constant term.
  • Assuming that all integrals can be evaluated directly without first simplifying or rewriting them into a standard form.
  • Overlooking the necessity to use integration by parts when faced with integrands that are products of functions, leading to more complicated integration attempts.