Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.5.6
Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)
Verified step by step guidance1
Start by factoring the denominator of the given expression \(\frac{z}{z^{3} - z^{2} - 6z}\). First, factor out the common factor \(z\):
\(z^{3} - z^{2} - 6z = z(z^{2} - z - 6)\).
Next, factor the quadratic expression \(z^{2} - z - 6\). Find two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\), so:
\(z^{2} - z - 6 = (z - 3)(z + 2)\).
Rewrite the original expression using the factored denominator:
\(\frac{z}{z(z - 3)(z + 2)}\).
Set up the partial fraction decomposition. Since the denominator factors are linear and distinct, express the fraction as:
\(\frac{z}{z(z - 3)(z + 2)} = \frac{A}{z} + \frac{B}{z - 3} + \frac{C}{z + 2}\), where \(A\), \(B\), and \(C\) are constants to be determined.
Multiply both sides of the equation by the common denominator \(z(z - 3)(z + 2)\) to clear the denominators:
\(z = A(z - 3)(z + 2) + B z (z + 2) + C z (z - 3)\).
This equation can now be expanded and simplified to solve for \(A\), \(B\), and \(C\) by equating coefficients or substituting convenient values of \(z\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 factoring the denominator and writing the original fraction as a sum of fractions with those factors as denominators.
Recommended video:
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into products of simpler polynomials. For partial fractions, factoring the denominator completely into linear or irreducible quadratic factors is essential to set up the decomposition correctly.
Recommended video:
07:00Taylor Polynomials
Solving Systems of Equations
After setting up the partial fractions, coefficients are found by equating numerators and solving the resulting system of linear equations. This step is crucial to determine the unknown constants in the decomposition.
Recommended video:
5:02Solving Logarithmic Equations
Related Practice
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((2x + 1)√(4x + 4x²)))
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y
Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)
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