Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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.
Recommended video:
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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.
Recommended video:
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
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)
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Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx
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