Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
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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 8.8.8
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
Guida verificata passo dopo passo1
Identify the integral to be evaluated: \(\int_0^1 \frac{dr}{r^{0.999}}\).
Rewrite the integrand using exponent rules: \(\frac{1}{r^{0.999}} = r^{-0.999}\), so the integral becomes \(\int_0^1 r^{-0.999} \, dr\).
Recall the power rule for integration: For \(\int r^n \, dr\), where \(n \neq -1\), the antiderivative is \(\frac{r^{n+1}}{n+1} + C\).
Apply the power rule to the integral: \(\int_0^1 r^{-0.999} \, dr = \left[ \frac{r^{-0.999 + 1}}{-0.999 + 1} \right]_0^1 = \left[ \frac{r^{0.001}}{0.001} \right]_0^1\).
Evaluate the definite integral by substituting the limits: calculate \(\frac{1^{0.001}}{0.001} - \frac{0^{0.001}}{0.001}\), noting the behavior of \(r^{0.001}\) as \(r\) approaches 0.
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Improper Integrals
Improper integrals involve integrands that are unbounded or have infinite limits. When the integrand approaches infinity at a boundary, the integral is evaluated as a limit to determine convergence or divergence.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Power Rule for Integration
The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. This rule is essential for integrating functions with variable exponents, allowing direct evaluation of integrals involving powers of the variable.
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Percorso guidato
04:04Power Rule for Indefinite Integrals
Convergence of Integrals with Singularities
When the integrand has a singularity (like r^(-0.999) near zero), determining if the integral converges requires analyzing the behavior near the singularity. If the integral's limit exists and is finite, the integral converges.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
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Evaluate the integrals in Exercises 39–54.
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Domanda del libro di testo
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
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