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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 8.5.4
Capitolo 8, Problema 8.5.4

Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)

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1
Recognize that the denominator \(x^{2} - 2x + 1\) can be factored. Factor it as a perfect square: \(x^{2} - 2x + 1 = (x - 1)^{2}\).
Set up the partial fraction decomposition for the rational expression \(\frac{2x + 2}{(x - 1)^{2}}\). Since the denominator is a repeated linear factor, express it as \(\frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\).
Multiply both sides of the equation by the denominator \((x - 1)^{2}\) to clear the fractions: \(2x + 2 = A(x - 1) + B\).
Expand the right side: \(2x + 2 = A x - A + B\). Then, group like terms: \(2x + 2 = A x + (B - A)\).
Equate the coefficients of corresponding powers of \(x\) on both sides to form a system of equations: For \(x\), \(2 = A\); for the constant term, \(2 = B - A\). Solve this system to find \(A\) and \(B\).

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex fraction into simpler terms with denominators that are factors of the original denominator.
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Partial Fraction Decomposition: Distinct Linear Factors

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials or identifying perfect square trinomials. Recognizing the factorization of the denominator is essential for setting up the partial fractions correctly.
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Partial Fraction Decomposition: Irreducible Quadratic Factors

Handling Repeated Factors in Denominators

When the denominator has repeated factors, such as (x - a)², partial fractions must include terms for each power of the repeated factor. This ensures the decomposition accounts for all components of the denominator for accurate expansion.
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Partial Fraction Decomposition: Repeated Linear Factors
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