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.
∫ dx / (x² √(x² + 1))
17
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 8.5.56
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
Guida verificata passo dopo passo1
Start by factoring the denominator \(x^{3} - 2x^{2} - 3x\). Factor out the common factor \(x\) first, then factor the quadratic expression that remains.
Rewrite the integral as \(\int \frac{x + 2}{x (x^{2} - 2x - 3)} \, dx\). Next, factor the quadratic \(x^{2} - 2x - 3\) into two binomials.
Express the integrand as a sum of partial fractions: \(\frac{x + 2}{x (x - 3)(x + 1)} = \frac{A}{x} + \frac{B}{x - 3} + \frac{C}{x + 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides by the denominator \(x (x - 3)(x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations for \(A\), \(B\), and \(C\).
Solve the system for \(A\), \(B\), and \(C\), then rewrite the integral as the sum of simpler integrals: \(\int \frac{A}{x} \, dx + \int \frac{B}{x - 3} \, dx + \int \frac{C}{x + 1} \, dx\). Integrate each term using the natural logarithm function.
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:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves factoring the denominator and expressing the integrand as a sum of simpler rational expressions. This method is especially useful when integrating rational functions where the degree of the numerator is less than the degree of the denominator.
Video consigliato:
Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its factors, such as linear or quadratic terms. This step is crucial before applying partial fraction decomposition because it reveals the denominator's roots and their multiplicities. For example, factoring the cubic denominator x³ - 2x² - 3x helps identify the terms needed for decomposition.
Video consigliato:
07:00Taylor Polynomials
Integration of Rational Functions
Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply standard integral formulas. After decomposition, each term corresponds to a basic integral form, like ∫1/(x - a) dx or ∫1/(x² + bx + c) dx, which can be integrated using logarithmic or inverse trigonometric functions.
Video consigliato:
6:04Intro to Rational Functions
Pratica correlata
Domanda del libro di testo
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx
29
views
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.
∫ (1 - r²)^(5/2) / r⁸ dr
28
views
Domanda del libro di testo
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx
4
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.
86. Find the volume of the solid generated by revolving the region about the x-axis.
22
views
Domanda del libro di testo
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
27
views