Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
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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 8.5.56
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
Verified step by step guidance1
Start by factoring the denominator \(x^{3} - 2x^{2} - 3x\). Factor out the common factor \(x\) first, then factor the quadratic expression that remains.
Rewrite the integral as \(\int \frac{x + 2}{x (x^{2} - 2x - 3)} \, dx\). Next, factor the quadratic \(x^{2} - 2x - 3\) into two binomials.
Express the integrand as a sum of partial fractions: \(\frac{x + 2}{x (x - 3)(x + 1)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{x + 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides by the denominator \(x (x - 3)(x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations for \(A\), \(B\), and \(C\).
Solve the system for \(A\), \(B\), and \(C\), then rewrite the integral as the sum of simpler integrals: \(\int \frac{A}{x} \, dx + \int \frac{B}{x - 3} \, dx + \int \frac{C}{x + 1} \, dx\). Integrate each term using the natural logarithm function.
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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 factoring the denominator and expressing the integrand as a sum of simpler rational expressions. This method is especially useful when integrating rational functions where the degree of the numerator is less than the degree of the denominator.
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Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its factors, such as linear or quadratic terms. This step is crucial before applying partial fraction decomposition because it reveals the denominator's roots and their multiplicities. For example, factoring the cubic denominator x³ - 2x² - 3x helps identify the terms needed for decomposition.
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07:00Taylor Polynomials
Integration of Rational Functions
Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply standard integral formulas. After decomposition, each term corresponds to a basic integral form, like ∫1/(x - a) dx or ∫1/(x² + bx + c) dx, which can be integrated using logarithmic or inverse trigonometric functions.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = c * x * √(25 - x²) over [0, 5]
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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.
∫ x² sin(x³) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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Textbook Question
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].