Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 1 to ∞ of ((ln z) / z) dz
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.10
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx
Verified step by step guidance1
Start by examining the integral: \(\int \frac{x}{x^{2} + 4x + 3} \, dx\). Notice that the denominator is a quadratic expression that can potentially be factored.
Factor the quadratic in the denominator: \(x^{2} + 4x + 3 = (x + 1)(x + 3)\). This will help in simplifying the integral or setting up partial fractions.
Since the degree of the numerator is less than the degree of the denominator, consider using partial fraction decomposition. Express \(\frac{x}{(x + 1)(x + 3)}\) as \(\frac{A}{x + 1} + \frac{B}{x + 3}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides by \((x + 1)(x + 3)\) to clear the denominators: \(x = A(x + 3) + B(x + 1)\). Expand and collect like terms to form an equation in terms of \(x\).
Equate the coefficients of like powers of \(x\) on both sides to solve for \(A\) and \(B\). Once \(A\) and \(B\) are found, rewrite the integral as the sum of two simpler integrals and integrate each term separately.
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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 a composite function or a function and its derivative.
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Substitution With an Extra Variable
Partial Fraction Decomposition
Partial fraction decomposition breaks down a rational function into simpler fractions that are easier to integrate. This method is applicable when the denominator can be factored into linear or quadratic terms. After decomposition, each simpler fraction can be integrated using basic integral formulas.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. Factoring the denominator, such as x² + 4x + 3, helps in simplifying the integral and applying partial fraction decomposition. Recognizing factorable quadratics is essential for breaking down complex rational expressions.
Recommended video:
07:00Taylor Polynomials
Related Practice
Textbook Question
Evaluate the integrals in Exercises 37–44.
∫ sec²(θ) sin³(θ) dθ
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ e^(ln√x) dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sin(2θ) dθ / (1 + cos(2θ))²
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
131. ∫ dx / (x√(1 − x⁴))
Textbook Question
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x sin(x) cos(x) dx