Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv
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.PE.26
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / (x (1 + ∛x))] dx
Verified step by step guidance1
Start by examining the integral \( \int \frac{1}{x (1 + \sqrt[3]{x})} \, dx \). Notice the presence of \( \sqrt[3]{x} \), which suggests a substitution involving \( x^{1/3} \).
Let \( t = \sqrt[3]{x} = x^{1/3} \). Then, express \( x \) in terms of \( t \): \( x = t^3 \).
Differentiate \( x = t^3 \) with respect to \( t \) to find \( dx \): \( dx = 3t^2 \, dt \).
Rewrite the integral in terms of \( t \) by substituting \( x = t^3 \), \( dx = 3t^2 \, dt \), and \( \sqrt[3]{x} = t \). The integral becomes \( \int \frac{1}{t^3 (1 + t)} \cdot 3t^2 \, dt \).
Simplify the integrand: \( \frac{3t^2}{t^3 (1 + t)} = \frac{3}{t (1 + t)} \). Now, the integral is \( \int \frac{3}{t (1 + t)} \, dt \). To integrate, use partial fraction decomposition on \( \frac{1}{t (1 + t)} \) and then integrate term-by-term.
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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 method 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 technique is especially useful when the integral contains composite functions or roots.
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Substitution With an Extra Variable
Properties of Exponents and Roots
Understanding how to manipulate exponents and roots is essential for rewriting expressions like ∛x (the cube root of x) as x^(1/3). This allows for easier algebraic manipulation and substitution during integration, enabling the integral to be expressed in terms of powers of x.
Recommended video:
06:21Properties of Functions
Integration of Rational Functions
Integrating rational functions involves integrating ratios of polynomials or expressions that can be transformed into such ratios. Techniques include partial fraction decomposition or substitution to simplify the integrand, making it easier to find the antiderivative.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
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Textbook Question
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Textbook Question
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