Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.5.15

Capitolo 8, Problema 8.5.15

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
15. x / ((x⁴ - 16)²)

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Recognize that the denominator is \( (x^4 - 16)^2 \), which is a repeated factor. First, factor \( x^4 - 16 \) as a difference of squares: \( x^4 - 16 = (x^2 - 4)(x^2 + 4) \).

Further factor \( x^2 - 4 \) as another difference of squares: \( x^2 - 4 = (x - 2)(x + 2) \). So the full factorization of the denominator is \( (x - 2)(x + 2)(x^2 + 4) \), and the entire denominator is \( [(x - 2)(x + 2)(x^2 + 4)]^2 \).

Since the denominator is squared, each factor appears with multiplicity 2. For partial fractions, include terms for each power from 1 up to 2 for each factor.

Set up the partial fraction decomposition as the sum of fractions with denominators \( (x - 2) \) and \( (x - 2)^2 \), \( (x + 2) \) and \( (x + 2)^2 \), and \( (x^2 + 4) \) and \( (x^2 + 4)^2 \). For linear factors, use constants in the numerators; for the irreducible quadratic \( x^2 + 4 \), use linear expressions in the numerator.

Write the decomposition as: \[ \frac{x}{(x^4 - 16)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{x + 2} + \frac{D}{(x + 2)^2} + \frac{Ex + F}{x^2 + 4} + \frac{Gx + H}{(x^2 + 4)^2} \] where \( A, B, C, D, E, F, G, H \) are constants to be determined.

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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 rational expression into a sum of fractions with simpler denominators, typically linear or quadratic factors.

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Partial Fraction Decomposition: Distinct Linear Factors

Factorization of Polynomials

Factorization involves expressing a polynomial as a product of its irreducible factors. For partial fractions, correctly factoring the denominator into linear and/or irreducible quadratic factors is essential to set up the decomposition properly.

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Taylor Polynomials

Handling Repeated Factors in Partial Fractions

When the denominator contains repeated factors, each power of the repeated factor must be included separately in the decomposition. For example, if (x⁴ - 16) is squared, terms with denominators (x⁴ - 16) and (x⁴ - 16)² must both appear with unknown constants.

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Percorso guidato

11:26

Partial Fraction Decomposition: Repeated Linear Factors

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