Domanda del libro di testo
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.
19
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 8.5.24
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx
Guida verificata passo dopo passo1
Recognize that the integrand is a rational function where the denominator is a repeated quadratic factor: \((4x^{2} + 1)^{2}\). Since the denominator is a repeated irreducible quadratic, set up the partial fraction decomposition accordingly.
Express the integrand as a sum of partial fractions of the form: \(\frac{8x^{2} + 8x + 2}{(4x^{2} + 1)^{2}} = \frac{Ax + B}{4x^{2} + 1} + \frac{Cx + D}{(4x^{2} + 1)^{2}}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
Multiply both sides of the equation by \((4x^{2} + 1)^{2}\) to clear the denominators, resulting in: \(8x^{2} + 8x + 2 = (Ax + B)(4x^{2} + 1) + (Cx + D)\).
Expand the right-hand side and collect like terms in powers of \(x\). Then, equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations for \(A\), \(B\), \(C\), and \(D\).
Solve the system of equations to find the values of \(A\), \(B\), \(C\), and \(D\). Once found, rewrite the integral as the sum of simpler integrals involving these partial fractions, which can then be integrated using standard techniques such as substitution and recognizing derivatives of inverse trigonometric functions.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
7mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is especially useful when integrating rational functions, as it breaks down complex expressions into manageable parts that can be integrated individually.
Video consigliato:
Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions often involves rewriting the integrand into simpler terms, such as polynomials or partial fractions. Recognizing when to use substitution, partial fractions, or other integration techniques is essential for solving these integrals effectively.
Video consigliato:
6:04Intro to Rational Functions
Handling Repeated Quadratic Factors
When the denominator contains repeated quadratic factors, partial fractions include terms with increasing powers of the quadratic in the denominator. Understanding how to set up and solve for coefficients in these cases is crucial for correctly decomposing and integrating the function.
Video consigliato:
Percorso guidato
13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Pratica correlata
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 4 to ∞ of (dx / (√x - 1))
5
views
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.
∫ (csc t sin 3t dt)
23
views
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.
∫ (dθ / cos θ - 1)
27
views
Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)
30
views
Domanda del libro di testo
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ cos(θ / 2) cos(7θ) dθ
25
views