Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
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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 8.5.18
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
Verified step by step guidance1
Recognize that the denominator \(x^{2} - 2x + 1\) can be factored as \((x - 1)^{2}\) since it is a perfect square trinomial.
Rewrite the integrand as \(\frac{x^{3}}{(x - 1)^{2}}\) and consider expressing it as a sum of partial fractions. Since the denominator is a repeated linear factor, the partial fraction decomposition will have the form \(\frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\).
Set up the equation: \(\frac{x^{3}}{(x - 1)^{2}} = \frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\). Multiply both sides by \((x - 1)^{2}\) to clear the denominators, resulting in \(x^{3} = A(x - 1) + B\).
Expand and simplify the right side: \(A(x - 1) + B = Ax - A + B\). Equate coefficients of powers of \(x\) on both sides to solve for constants \(A\) and \(B\).
Once \(A\) and \(B\) are found, rewrite the integrand as the sum of partial fractions and integrate term-by-term over the interval from \(-1\) to \(0\). Remember to apply the definite integral limits after integrating.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, often linear or quadratic factors. This method is especially useful when the denominator can be factored into repeated or distinct linear terms.
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Partial Fraction Decomposition: Distinct Linear Factors
Polynomial Division
Polynomial division is used when the degree of the numerator is equal to or greater than the degree of the denominator. It simplifies the integrand by dividing the polynomials to rewrite the integrand as a polynomial plus a proper fraction. This step is essential before applying partial fraction decomposition if the integrand is an improper rational function.
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Definite Integration
Definite integration calculates the exact area under a curve between two limits. After simplifying the integrand, the integral is evaluated by finding the antiderivative and then applying the Fundamental Theorem of Calculus to compute the difference between the values at the upper and lower bounds. This process yields a numerical value representing the integral over the specified interval.
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Related Practice
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
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x⁴ - 1) / (x⁵ - 5x + 1) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-2x) sin(2x) dx
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
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.
∫ (dx / ((x - 2)√(x² - 4x + 3)))
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