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If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) 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 8.5.12
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.
12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))
Guida verificata passo dopo passo1
Identify the factors in the denominator: \(x^{2} - 8x + 16\) and \(x^{2} + 3x + 4\). Notice that \(x^{2} - 8x + 16\) is a perfect square trinomial and can be factored as \((x - 4)^{2}\). The other quadratic \(x^{2} + 3x + 4\) does not factor nicely over the reals, so it remains as is.
Since \((x - 4)^{2}\) is a repeated linear factor, set up terms for both the first power and the second power of \((x - 4)\) in the decomposition. For each linear factor, the numerator is a constant: \(\frac{A}{x - 4} + \frac{B}{(x - 4)^{2}}\).
For the irreducible quadratic factor \(x^{2} + 3x + 4\), the numerator must be a linear expression of the form $Cx + D$ because the degree of the numerator is one less than the degree of the denominator factor.
Combine all parts to write the partial fraction decomposition setup as: \(\frac{A}{x - 4} + \frac{B}{(x - 4)^{2}} + \frac{Cx + D}{x^{2} + 3x + 4}\).
This setup is the appropriate form for the partial fraction decomposition of the given expression. The next step (not required here) would be to multiply both sides by the denominator and solve for the constants \(A\), \(B\), \(C\), and \(D\).
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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, making integration or other operations easier. It involves breaking down a complex fraction into a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.
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10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of simpler polynomials. Recognizing perfect square trinomials or irreducible quadratics is essential to determine the correct form of partial fractions, especially when denominators contain quadratic factors.
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Percorso guidato
13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Form of Partial Fractions for Quadratic Denominators
When the denominator includes irreducible quadratic factors, the corresponding partial fractions have linear numerators (Ax + B) over these quadratics. For repeated or distinct quadratic factors, each term in the decomposition must reflect the degree and multiplicity of the factor.
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13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Pratica correlata
Domanda del libro di testo
7–84. Evaluate the following integrals.
64. ∫ (ln(ax))/x dx, where a ≠ 0
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Domanda del libro di testo
79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
82. ∫ (sin⁻¹(ax)) / x² dx, a > 0
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Domanda del libro di testo
94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.
For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?
What is its value when it exists?
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Domanda del libro di testo
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)
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7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
23. ∫ 1/(25 - x²)^(3/2) dx
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