Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ sin(θ) dθ / (cos²θ + cos θ - 2)
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 8.5.12
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx
Verified step by step guidance1
First, factor the quadratic denominator \(x^{2} - 7x + 12\). Find two numbers that multiply to 12 and add to -7. This factors as \((x - 3)(x - 4)\).
Express the integrand as a sum of partial fractions: write \(\frac{2x + 1}{(x - 3)(x - 4)} = \frac{A}{x - 3} + \frac{B}{x - 4}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides of the equation by the denominator \((x - 3)(x - 4)\) to clear the fractions: \(2x + 1 = A(x - 4) + B(x - 3)\).
Expand the right side and collect like terms: \(2x + 1 = A x - 4A + B x - 3B = (A + B) x + (-4A - 3B)\). Then, equate the coefficients of \(x\) and the constant terms on both sides to form a system of equations: \(2 = A + B\) and \(1 = -4A - 3B\).
Solve the system of equations for \(A\) and \(B\). Once you find \(A\) and \(B\), rewrite the integral as \(\int \frac{A}{x - 3} dx + \int \frac{B}{x - 4} dx\) and integrate each term separately using the natural logarithm rule for integrals of the form \(\int \frac{1}{x - c} dx\).
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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 break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.
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Partial Fraction Decomposition: Distinct Linear Factors
Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting a quadratic polynomial as a product of two binomials. This step is essential for partial fraction decomposition because it identifies the simpler linear factors in the denominator needed to split the integrand.
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13:42Partial Fraction Decomposition: Irreducible Quadratic Factors
Integration of Rational Functions
Integration of rational functions often requires rewriting the integrand into simpler parts, such as partial fractions, to apply basic integration rules. Once decomposed, each term can be integrated using standard formulas, such as the integral of 1/(x - a) being ln|x - a| + C.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
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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 0 to 1 of (dt / (t - sin t))
(Hint: t ≥ sin t for t ≥ 0)
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Textbook Question
Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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Textbook Question
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ