Domanda del libro di testo
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
48. ∫ √(9 - 4x²) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.5.9
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.
9. 4/(x⁵ - 5x³ + 4x)
Guida verificata passo dopo passo1
First, factor the denominator completely. The denominator is \(x^{5} - 5x^{3} + 4x\). Start by factoring out the common factor \(x\):
\[x^{5} - 5x^{3} + 4x = x(x^{4} - 5x^{2} + 4)\]
Next, factor the quartic polynomial \(x^{4} - 5x^{2} + 4\). Treat \(x^{2}\) as a variable, say \(y = x^{2}\), so the expression becomes \(y^{2} - 5y + 4\). Factor this quadratic:
\[y^{2} - 5y + 4 = (y - 4)(y - 1)\]
Substitute back \(y = x^{2}\) to get:
\[ (x^{2} - 4)(x^{2} - 1) \]
Further factor the difference of squares:
\[x^{2} - 4 = (x - 2)(x + 2)\]
\[x^{2} - 1 = (x - 1)(x + 1)\]
So the full factorization of the denominator is:
\[x(x - 2)(x + 2)(x - 1)(x + 1)\]
Since all factors are linear and distinct, the partial fraction decomposition will have terms of the form:
\[\frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} + \frac{D}{x - 1} + \frac{E}{x + 1}\]
where \(A\), \(B\), \(C\), \(D\), and \(E\) are constants to be determined.
Set up the equation:
\[\frac{4}{x^{5} - 5x^{3} + 4x} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2} + \frac{D}{x - 1} + \frac{E}{x + 1}\]
This is the appropriate form of the partial fraction decomposition without solving for the constants.
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Partial Fraction Decomposition
Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other operations by breaking down complex expressions into manageable parts.
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Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of its irreducible factors. For partial fractions, factoring the denominator completely into linear and/or irreducible quadratic factors is essential to determine the form of the decomposition.
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07:00Taylor Polynomials
Form of Partial Fractions for Repeated and Higher-Degree Factors
When the denominator has repeated linear factors or higher-degree factors, the partial fraction decomposition includes terms for each power of the repeated factor and numerators of appropriate degree for irreducible quadratic factors. Setting up the correct form is crucial before solving for unknown constants.
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Percorso guidato
11:26Partial Fraction Decomposition: Repeated Linear Factors
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River flow rates
The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and ∫(0 to 24) r(t) dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate ∫(0 to 24) r(t) dt using the Trapezoidal Rule and Simpson's Rule with the following values of n.
n = 6
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