Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.5.42
Evaluate the integrals in Exercises 39–54.
∫ sin(θ) dθ / (cos²θ + cos θ - 2)
Verified step by step guidance1
Start by examining the integral: \(\int \frac{\sin(\theta)}{\cos^{2}(\theta) + \cos(\theta) - 2} \, d\theta\). Notice that the denominator is a quadratic expression in terms of \(\cos(\theta)\).
Let \(u = \cos(\theta)\). Then, compute \(du = -\sin(\theta) \, d\theta\), which implies \(-du = \sin(\theta) \, d\theta\). This substitution will help simplify the integral.
Rewrite the integral in terms of \(u\): replace \(\sin(\theta) \, d\theta\) with \(-du\), and the denominator becomes \(u^{2} + u - 2\). So the integral becomes \(\int \frac{-du}{u^{2} + u - 2}\).
Factor the quadratic in the denominator: \(u^{2} + u - 2 = (u + 2)(u - 1)\). This allows you to use partial fraction decomposition to express \(\frac{1}{(u + 2)(u - 1)}\) as a sum of simpler fractions.
Set up the partial fractions: \(\frac{1}{(u + 2)(u - 1)} = \frac{A}{u + 2} + \frac{B}{u - 1}\). Solve for constants \(A\) and \(B\), then integrate each term separately with respect to \(u\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Trigonometric Functions
This involves techniques to integrate functions containing sine, cosine, and other trigonometric expressions. Recognizing patterns and using identities can simplify the integral, making it easier to solve.
Recommended video:
6:04
Introduction to Trigonometric Functions
Trigonometric Identities and Factorization
Using identities like factoring quadratic expressions in terms of cosine helps simplify the denominator. For example, factoring cos²θ + cosθ - 2 into (cosθ + 2)(cosθ - 1) can make the integral more manageable.
Recommended video:
7:17Verifying Trig Equations as Identities
Substitution Method in Integration
Substitution involves changing variables to simplify the integral. Here, letting u = cosθ transforms the integral into a rational function in u, which is easier to integrate.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
Textbook Question
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² dx) / ((x - 1)(x² + 2x + 1))
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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
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx