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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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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.5.42
Evaluate the integrals in Exercises 39–54.
∫ sin(θ) dθ / (cos²θ + cos θ - 2)
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
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.
Video consigliato:
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.
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07:33Euler's Method
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