Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (x^2 + 6x) / (x^2 + 3)^2 dx
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 8.5.8
Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)
Verified step by step guidance1
First, factor the denominator \(t^4 + 9t^2\) by taking out the common factor \(t^2\), so it becomes \(t^2(t^2 + 9)\).
Set up the partial fraction decomposition for the expression \(\frac{t^4 + 9}{t^2(t^2 + 9)}\). Since \(t^2\) is a repeated linear factor, and \(t^2 + 9\) is an irreducible quadratic, write it as: \(\frac{A}{t} + \frac{B}{t^2} + \frac{Ct + D}{t^2 + 9}\).
Multiply both sides of the equation by the denominator \(t^2(t^2 + 9)\) to clear the fractions, resulting in: \(t^4 + 9 = A t (t^2 + 9) + B (t^2 + 9) + (Ct + D) t^2\).
Expand the right-hand side and collect like terms in powers of \(t\) to form a polynomial equation: \(t^4 + 9 = A t^3 + 9 A t + B t^2 + 9 B + C t^3 + D t^2\).
Equate the coefficients of corresponding powers of \(t\) on both sides to form a system of equations, then solve for \(A\), \(B\), \(C\), and \(D\).
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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 technique used to express a rational function as a sum of simpler fractions, making integration or simplification easier. It involves factoring the denominator and writing the original fraction as a sum of fractions with those factors as denominators.
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Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials involves 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 correct form of the decomposition.
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Handling Repeated and Quadratic Factors
When the denominator contains repeated factors or irreducible quadratic factors, the partial fraction setup must include terms for each power of the repeated factor and linear numerators for quadratic factors. This ensures the decomposition accounts for all components of the denominator.
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13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Related Practice
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ √(y² - 25) / y³ dy, where y > 5
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / (x² + 3x)) dx
1
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Textbook Question
Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
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