Domanda del libro di testo
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.5.62
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int \frac{2}{x(\ln x - 2)^3} \, dx\).
Recognize that the integral involves a composite function \(\ln x - 2\) in the denominator, suggesting a substitution related to \(\ln x\).
Let \(u = \ln x - 2\). Then, compute the differential \(du\): since \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have \(du = \frac{1}{x} dx\).
Rewrite the integral in terms of \(u\) and \(du\). Notice that \(dx = x \, du\), so substitute accordingly to express the integral fully in terms of \(u\).
After substitution, the integral becomes \(\int \frac{2}{u^3} \, du\). Now, integrate this expression using the power rule for integrals.
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Integration by Substitution
Integration by substitution is a technique 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 method is especially useful when the integral contains a composite function, such as ln(x) in this problem.
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04:27
Substitution With an Extra Variable
Properties of Logarithmic Functions
Understanding the properties of logarithmic functions, like ln(x), is essential for manipulating and simplifying expressions involving logs. Recognizing how derivatives of ln(x) relate to 1/x helps in choosing appropriate substitutions and solving integrals involving logarithmic terms.
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Percorso guidato
06:21Properties of Functions
Handling Rational Functions in Integration
Rational functions are ratios of polynomials or expressions involving variables in numerator and denominator. Integrating rational functions often requires algebraic manipulation or substitution to simplify the integrand. In this problem, the presence of x in the denominator combined with a function of ln(x) suggests careful handling of the rational expression.
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6:04Intro to Rational Functions
Pratica correlata
Domanda del libro di testo
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx
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Domanda del libro di testo
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.
∫ (√x / (1 + x³)) dx
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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.
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Domanda del libro di testo
What is the largest value that
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Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ x √(x² - 4) dx
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