Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(2θ) cos³(2θ) dθ
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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 8.5.60
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx
Verified step by step guidance1
First, examine the integral \( \int \frac{x^{4} - 1}{x^{5} - 5x + 1} \, dx \) and consider if the numerator is related to the derivative of the denominator. This is a common strategy for integrals involving rational functions.
Compute the derivative of the denominator: \( \frac{d}{dx} (x^{5} - 5x + 1) = 5x^{4} - 5 \).
Notice that the numerator \( x^{4} - 1 \) is similar to \( \frac{1}{5} \) times the derivative of the denominator, since \( 5x^{4} - 5 = 5(x^{4} - 1) \). This suggests rewriting the integral in terms of \( \frac{f'(x)}{f(x)} \) where \( f(x) = x^{5} - 5x + 1 \).
Rewrite the integral as \( \int \frac{x^{4} - 1}{x^{5} - 5x + 1} \, dx = \int \frac{1}{5} \cdot \frac{5x^{4} - 5}{x^{5} - 5x + 1} \, dx = \frac{1}{5} \int \frac{f'(x)}{f(x)} \, dx \).
Use the formula \( \int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C \) to express the integral in terms of the natural logarithm of the denominator, multiplied by the constant factor \( \frac{1}{5} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques include various methods such as substitution, partial fractions, and integration by parts, used to evaluate integrals that are not straightforward. Choosing the right technique depends on the form of the integrand and can simplify complex expressions into manageable parts.
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Integration by Parts for Definite Integrals
Polynomial Division and Simplification
When integrating rational functions, polynomial division helps simplify the integrand if the numerator's degree is equal to or higher than the denominator's. Simplifying the expression can make the integral easier to evaluate by breaking it into simpler terms or revealing a suitable substitution.
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07:00Taylor Polynomials
Substitution Method
The substitution method involves changing variables to simplify the integral, often by letting u equal a function inside the integrand. This technique is especially useful when the derivative of the chosen substitution appears elsewhere in the integrand, enabling a straightforward integration.
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07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 3sin(x/3) dx
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫₀¹/√2 2x arcsin(x²) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-2x) sin(2x) dx
Textbook Question
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x³ dx) / (x² - 2x + 1) from -1 to 0
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Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x² - 4) / x dx