Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
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 7.6.75
Evaluate the integrals in Exercises 53–76.
75. ∫y dy/√(1-y^4)
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{y \, dy}{\sqrt{1 - y^{4}}}\).
Consider a substitution to simplify the expression under the square root. Notice that the denominator involves \(1 - y^{4}\), which can be rewritten as \(1 - (y^{2})^{2}\), suggesting a substitution involving \(y^{2}\).
Let \(u = y^{2}\). Then, compute \(du\) in terms of \(dy\): since \(u = y^{2}\), we have \(du = 2y \, dy\), or equivalently, \(y \, dy = \frac{du}{2}\).
Rewrite the integral in terms of \(u\): substitute \(y \, dy\) with \(\frac{du}{2}\) and \(\sqrt{1 - y^{4}}\) with \(\sqrt{1 - u^{2}}\), so the integral becomes \(\int \frac{\frac{du}{2}}{\sqrt{1 - u^{2}}} = \frac{1}{2} \int \frac{du}{\sqrt{1 - u^{2}}}\).
Recognize that \(\int \frac{du}{\sqrt{1 - u^{2}}}\) is a standard integral whose antiderivative is \(\arcsin(u) + C\). After integrating, substitute back \(u = y^{2}\) to express the answer in terms of \(y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques involve methods such as substitution, integration by parts, and recognizing standard integral forms. For the given integral, substitution is often used to simplify the integrand into a more manageable form, enabling direct integration.
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Substitution Method
The substitution method replaces a complicated expression with a single variable to simplify the integral. In this problem, setting a substitution like u = y² or u = y⁴ can transform the integral into a standard form, making it easier to evaluate.
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Integrals Involving Radical Expressions
Integrals containing radicals, such as √(1 - y⁴), often require recognizing patterns or using trigonometric or hyperbolic substitutions. Understanding how to handle these radicals is essential to rewrite the integral into a solvable form.
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Related Practice
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Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x⁵