Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.5.12
Capitolo 8, Problema 8.5.12

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx

Guida verificata passo dopo passo
1
First, factor the quadratic denominator \(x^{2} - 7x + 12\). Find two numbers that multiply to 12 and add to -7. This factors as \((x - 3)(x - 4)\).
Express the integrand as a sum of partial fractions: write \(\frac{2x + 1}{(x - 3)(x - 4)} = \frac{A}{x - 3} + \frac{B}{x - 4}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the denominator \((x - 3)(x - 4)\) to clear the fractions: \(2x + 1 = A(x - 4) + B(x - 3)\).
Expand the right side and collect like terms: \(2x + 1 = A x - 4A + B x - 3B = (A + B) x + (-4A - 3B)\). Then, equate the coefficients of \(x\) and the constant terms on both sides to form a system of equations: \(2 = A + B\) and \(1 = -4A - 3B\).
Solve the system of equations for \(A\) and \(B\). Once you find \(A\) and \(B\), rewrite the integral as \(\int \frac{A}{x - 3} dx + \int \frac{B}{x - 4} dx\) and integrate each term separately using the natural logarithm rule for integrals of the form \(\int \frac{1}{x - c} dx\).

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:
4m

Concetti 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 expressing the integrand as a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.
Video consigliato:
Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. This step is essential for partial fraction decomposition because it identifies the simpler linear factors in the denominator needed to split the integrand.
Video consigliato:
Percorso guidato
13:42
Partial Fraction Decomposition: Irreducible Quadratic Factors

Integration of Rational Functions

Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply basic integration rules. Once decomposed, each term can be integrated using standard formulas, such as the integral of 1/(x - a) being ln|x - a| + C.
Video consigliato:
6:04
Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = (1/x) over [c, c + 1]

49
views
Domanda del libro di testo

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ 1 / √(x² - 2x + 5) dx

26
views
Domanda del libro di testo

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² dx) / ((x - 1)(x² + 2x + 1))

31
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 0 to 1 of (dt / (t - sin t))

(Hint: t ≥ sin t for t ≥ 0)

23
views
Domanda del libro di testo

Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].

41
views
Domanda del libro di testo

Evaluate the integrals in Exercises 1–22.

∫ cos³(4x) dx

25
views