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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.5.2
Chapter 8, Problem 8.5.2

Expand the quotients in Exercises 1–8 by partial fractions.
(5x - 7) / (x² - 3x + 2)

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First, factor the denominator \(x^{2} - 3x + 2\) into its linear factors. To do this, find two numbers that multiply to \(2\) and add to \(-3\).
Rewrite the original expression as a sum of partial fractions with unknown coefficients: \(\frac{5x - 7}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b}\), where \(a\) and \(b\) are the roots found from factoring.
Multiply both sides of the equation by the denominator \((x - a)(x - b)\) to clear the fractions, resulting in an equation involving polynomials: \(5x - 7 = A(x - b) + B(x - a)\).
Expand the right-hand side and collect like terms in \(x\). This will give you an equation of the form \(5x - 7 = (A + B)x - (Ab + Ba)\).
Equate the coefficients of corresponding powers of \(x\) on both sides to form a system of linear equations. Solve this system to find the values of \(A\) and \(B\).

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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 with linear or quadratic denominators. This technique simplifies integration and other operations by breaking down complex fractions into manageable parts.
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Partial Fraction Decomposition: Distinct Linear Factors

Factoring Quadratic Denominators

Factoring the quadratic denominator into linear factors is essential for partial fraction decomposition. For example, x² - 3x + 2 factors into (x - 1)(x - 2), allowing the original fraction to be split into terms with these simpler denominators.
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Partial Fraction Decomposition: Irreducible Quadratic Factors

Setting Up and Solving Systems of Equations

After expressing the fraction as a sum of partial fractions with unknown coefficients, you multiply both sides by the denominator and equate coefficients of like terms. This process yields a system of linear equations to solve for the unknown constants.
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Solving Logarithmic Equations
Related Practice
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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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Textbook Question

Use any method to evaluate the integrals in Exercises 55–66.

∫ x² √(1 - x²) dx

Textbook Question

Average value

In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.

Find the average value of y over the interval 0 ≤ t ≤ 2π.

Textbook Question

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

Textbook Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.

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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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