Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
47. ∫(1/r)csc²(1+ln(r))dr
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Ch. 7 - Transcendental Functions
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 7, Problema 7.PE.125
In Exercises 125–128 solve the differential equation.
125. dy/dx = √y cos(√y)
Guida verificata passo dopo passo1
Rewrite the differential equation \( \frac{dy}{dx} = \sqrt{y} \cos(\sqrt{y}) \) to separate variables. Express it as \( \frac{dy}{\sqrt{y} \cos(\sqrt{y})} = dx \).
Make the substitution \( u = \sqrt{y} \), which implies \( y = u^2 \) and \( dy = 2u \, du \).
Rewrite the left side integral in terms of \( u \): \( \frac{dy}{\sqrt{y} \cos(\sqrt{y})} = \frac{2u \, du}{u \cos(u)} = \frac{2 \, du}{\cos(u)} \).
Set up the integral \( \int \frac{2}{\cos(u)} \, du = \int dx \), which simplifies to \( 2 \int \sec(u) \, du = x + C \), where \( C \) is the constant of integration.
Integrate \( \sec(u) \) using the standard formula \( \int \sec(u) \, du = \ln | \sec(u) + \tan(u) | + C \), then substitute back \( u = \sqrt{y} \) to express the solution implicitly.
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Separable Differential Equations
A separable differential equation can be written as a product of a function of y and a function of x, allowing variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the solution.
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Solving Separable Differential Equations
Substitution Method
Substitution involves introducing a new variable to simplify the differential equation, especially when the equation contains composite functions like √y. For example, letting u = √y can transform the equation into a more manageable form for integration.
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07:33Euler's Method
Integration Techniques
Solving the separated equation requires integrating functions involving trigonometric and algebraic expressions. Familiarity with integrating functions like cos(u) and handling integrals after substitution is essential to find the explicit or implicit solution.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
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