Domanda del libro di testo
Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.9.19
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx
Guida verificata passo dopo passo1
First, analyze the integral \( \int_{1}^{\infty} \frac{3x^{2} + 1}{x^{3} + x} \, dx \) to determine if it is an improper integral due to the infinite upper limit.
Next, simplify the integrand by factoring the denominator: \( x^{3} + x = x(x^{2} + 1) \). Rewrite the integrand as \( \frac{3x^{2} + 1}{x(x^{2} + 1)} \).
Use partial fraction decomposition to express \( \frac{3x^{2} + 1}{x(x^{2} + 1)} \) in the form \( \frac{A}{x} + \frac{Bx + C}{x^{2} + 1} \). Set up the equation and solve for constants \( A, B, \) and \( C \).
Rewrite the integral as the sum of simpler integrals: \( \int_{1}^{\infty} \frac{A}{x} \, dx + \int_{1}^{\infty} \frac{Bx + C}{x^{2} + 1} \, dx \).
Evaluate each integral separately by taking the limit as the upper bound approaches infinity. Determine if each integral converges or diverges, and combine the results to conclude about the original integral.
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Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, we take limits approaching the problematic point, such as infinity, to determine if the integral converges to a finite value or diverges.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Integration of Rational Functions
Rational functions are ratios of polynomials. Integrating them often requires techniques like partial fraction decomposition to rewrite the integrand into simpler terms that can be integrated individually.
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6:04Intro to Rational Functions
Convergence Tests for Improper Integrals
To determine if an improper integral converges, we analyze the behavior of the integrand as the variable approaches infinity or a discontinuity. Comparison tests or limit evaluations help decide whether the integral has a finite value or diverges.
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Percorso guidato
11:11Improper Integrals: Infinite Intervals
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