Domanda del libro di testo
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
58
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 8.5.6
Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)
Guida verificata passo dopo passo1
Start by factoring the denominator of the given expression \(\frac{z}{z^{3} - z^{2} - 6z}\). First, factor out the common factor \(z\):
\(z^{3} - z^{2} - 6z = z(z^{2} - z - 6)\).
Next, factor the quadratic expression \(z^{2} - z - 6\). Find two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so:
\(z^{2} - z - 6 = (z - 3)(z + 2)\).
Rewrite the original expression using the factored denominator:
\(\frac{z}{z(z - 3)(z + 2)}\).
Set up the partial fraction decomposition. Since the denominator factors are linear and distinct, express the fraction as:
\(\frac{z}{z(z - 3)(z + 2)} = \frac{A}{z} + \frac{B}{z - 3} + \frac{C}{z + 2}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides of the equation by the common denominator \(z(z - 3)(z + 2)\) to clear the denominators:
\(z = A(z - 3)(z + 2) + B z (z + 2) + C z (z - 3)\).
This equation can now be expanded and simplified to solve for \(A\), \(B\), and \(C\) by equating coefficients or substituting convenient values of \(z\).
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:
5mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves factoring the denominator and writing the original fraction as a sum of fractions with those factors as denominators.
Video consigliato:
Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into products of simpler polynomials. For partial fractions, factoring the denominator completely into linear or irreducible quadratic factors is essential to set up the decomposition correctly.
Video consigliato:
07:00Taylor Polynomials
Solving Systems of Equations
After setting up the partial fractions, coefficients are found by equating numerators and solving the resulting system of linear equations. This step is crucial to determine the unknown constants in the decomposition.
Video consigliato:
5:02Solving Logarithmic Equations
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx
32
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.
∫ (dx / ((2x + 1)√(4x + 4x²)))
30
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.
∫ (2^(√y) dy) / 2√y
30
views
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.
83. Find the area of the region.
21
views
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 π to ∞ of ((1 + sin x) / x² dx)
25
views