Textbook Question
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
Ch. 7 - Transcendental Functions
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 7, Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Verified step by step guidance1
Identify the integral to evaluate: \(\int_{-1}^{-\frac{\sqrt{2}}{2}} \frac{dy}{y \sqrt{4y^{2} - 1}}\).
Recognize that the integrand involves a square root of a quadratic expression, \(\sqrt{4y^{2} - 1}\), which suggests a trigonometric substitution to simplify the square root.
Use the substitution \(y = \frac{1}{2} \sec(\theta)\), because \(4y^{2} - 1 = 4 \left( \frac{1}{2} \sec(\theta) \right)^{2} - 1 = \sec^{2}(\theta) - 1 = \tan^{2}(\theta)\), which simplifies the square root to \(\tan(\theta)\).
Compute \(dy\) in terms of \(d\theta\): \(dy = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta\), and rewrite the integral entirely in terms of \(\theta\).
Change the limits of integration from \(y\) to \(\theta\) by solving \(y = \frac{1}{2} \sec(\theta)\) for the given bounds, then integrate the resulting expression with respect to \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals and Limits of Integration
A definite integral calculates the net area under a curve between two specific points, called limits of integration. Understanding how to apply these limits correctly is essential for evaluating the integral's numerical value over the given interval.
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Definition of the Definite Integral
Integration Techniques for Rational Functions Involving Roots
Integrals involving expressions like 1/(y√(4y² - 1)) often require substitution methods or recognizing standard integral forms. Techniques such as trigonometric substitution or algebraic manipulation simplify the integrand to a more manageable form.
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Domain and Validity of the Integrand
Before integrating, it is crucial to analyze the domain where the integrand is defined, especially when involving square roots and denominators. Ensuring the expression under the root is non-negative and the denominator is non-zero avoids invalid or undefined values.
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Related Practice
Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
Textbook Question
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Textbook Question
Evaluate the integrals in Exercises 41–60.
55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ
1
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
25. y = sinh⁻¹(√x)