Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.5.62
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{2}{x(\ln x - 2)^3} \, dx\).
Recognize that the integral involves a composite function \(\ln x - 2\) in the denominator, suggesting a substitution related to \(\ln x\).
Let \(u = \ln x - 2\). Then, compute the differential \(du\): since \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have \(du = \frac{1}{x} dx\).
Rewrite the integral in terms of \(u\) and \(du\). Notice that \(dx = x \, du\), so substitute accordingly to express the integral fully in terms of \(u\).
After substitution, the integral becomes \(\int \frac{2}{u^3} \, du\). Now, integrate this expression using the power rule for integrals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a technique used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This method is especially useful when the integral contains a composite function, such as ln(x) in this problem.
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Substitution With an Extra Variable
Properties of Logarithmic Functions
Understanding the properties of logarithmic functions, like ln(x), is essential for manipulating and simplifying expressions involving logs. Recognizing how derivatives of ln(x) relate to 1/x helps in choosing appropriate substitutions and solving integrals involving logarithmic terms.
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06:21Properties of Functions
Handling Rational Functions in Integration
Rational functions are ratios of polynomials or expressions involving variables in numerator and denominator. Integrating rational functions often requires algebraic manipulation or substitution to simplify the integrand. In this problem, the presence of x in the denominator combined with a function of ln(x) suggests careful handling of the rational expression.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx
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Textbook Question
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
Textbook Question
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx