Textbook Question
10 10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
k = 1 k = 1
10
c. Σ (aₖ + bₖ - 1)
k = 1
1
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Ch. 5 - Integrals
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 5, Problem 5.PE.45
Evaluate the integrals in Exercises 37–46.
∫(sin 2θ - cos 2θ)/(sin 2θ + cos 2θ)³dθ
Verified step by step guidance1
Start by examining the integral \( \int \frac{\sin 2\theta - \cos 2\theta}{(\sin 2\theta + \cos 2\theta)^3} \, d\theta \). Notice that the numerator and denominator involve \( \sin 2\theta + \cos 2\theta \) and its derivative might be related to the numerator.
Set \( u = \sin 2\theta + \cos 2\theta \). Then, compute \( \frac{du}{d\theta} \) to find the differential \( du \). Since \( \frac{d}{d\theta}(\sin 2\theta) = 2\cos 2\theta \) and \( \frac{d}{d\theta}(\cos 2\theta) = -2\sin 2\theta \), we get \( \frac{du}{d\theta} = 2\cos 2\theta - 2\sin 2\theta = 2(\cos 2\theta - \sin 2\theta) \).
Rewrite the numerator \( \sin 2\theta - \cos 2\theta \) in terms of \( \cos 2\theta - \sin 2\theta \) to relate it to \( du/d\theta \). Notice that \( \sin 2\theta - \cos 2\theta = - (\cos 2\theta - \sin 2\theta) \).
Express the integral in terms of \( u \) and \( du \) by substituting the numerator and denominator accordingly. The integral becomes \( \int \frac{- (\cos 2\theta - \sin 2\theta)}{u^3} \, d\theta \). Using the expression for \( du \), solve for \( d\theta \) and substitute into the integral.
Simplify the integral to a function of \( u \) and \( du \), which should be easier to integrate. After integration, substitute back \( u = \sin 2\theta + \cos 2\theta \) to express the answer in terms of \( \theta \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They help simplify expressions, such as rewriting sin 2θ and cos 2θ in terms of other functions or combining terms. Recognizing and applying these identities is essential to simplify the integral's numerator and denominator.
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Verifying Trig Equations as Identities
Substitution Method in Integration
The substitution method involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral (e.g., u = sin 2θ + cos 2θ), the integral can be transformed into a more manageable form. This technique is crucial for integrals where the derivative of the substitution appears elsewhere in the integrand.
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Integration of Rational Functions
Integration of rational functions involves integrating ratios of polynomials or expressions raised to powers. When the integrand is a rational function of trigonometric expressions, rewriting it and using substitution can reduce it to a standard form. Understanding how to handle powers in the denominator and simplify the integrand is key to solving such integrals.
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