Domanda del libro di testo
23-64. Integration Evaluate the following integrals.
41. ∫₋₁¹ x/(x + 3)² 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 8.5.54
23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz
Guida verificata passo dopo passo1
Start by expressing the integrand \( \frac{z + 1}{z(z^2 + 4)} \) as a sum of partial fractions. Since the denominator factors into \( z \) and \( z^2 + 4 \), set up the decomposition as \( \frac{z + 1}{z(z^2 + 4)} = \frac{A}{z} + \frac{Bz + C}{z^2 + 4} \).
Multiply both sides of the equation by the denominator \( z(z^2 + 4) \) to clear the fractions, resulting in \( z + 1 = A(z^2 + 4) + (Bz + C)z \).
Expand the right-hand side to get \( z + 1 = A z^2 + 4A + B z^2 + C z \), then group like terms: \( (A + B) z^2 + C z + 4A \).
Equate the coefficients of corresponding powers of \( z \) on both sides: For \( z^2 \), \( A + B = 0 \); for \( z \), \( C = 1 \); and for the constant term, \( 4A = 1 \). Solve this system to find \( A, B, \) and \( C \).
Rewrite the integral using the partial fractions found, then integrate each term separately: \( \int \frac{A}{z} dz + \int \frac{Bz + C}{z^2 + 4} dz \). Use standard integral formulas such as \( \int \frac{1}{z} dz = \ln|z| + C \) and for the second integral, consider splitting it into two integrals and using substitution or recognizing the form \( \int \frac{z}{z^2 + a^2} dz \) and \( \int \frac{1}{z^2 + a^2} dz \).
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Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. It involves expressing a rational function as a sum of simpler terms with linear or quadratic denominators, facilitating straightforward integration.
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Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions often requires rewriting the integrand into simpler parts, such as polynomials or partial fractions. Recognizing standard integral forms and applying appropriate methods like substitution or partial fractions is essential for solving these integrals.
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6:04Intro to Rational Functions
Handling Quadratic Denominators
When the denominator includes irreducible quadratic factors, integration may involve logarithmic or inverse trigonometric functions. Understanding how to integrate terms with quadratic denominators, often after partial fraction decomposition, is crucial for evaluating such integrals.
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06:11Limits of Rational Functions: Denominator = 0
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