Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ ln(x + x²) dx
Ch. 8 - Techniques of Integration
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 8, Problem 8.4.54
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
Verified step by step guidance1
Rewrite the given differential equation in the form \( \frac{dy}{dx} = \frac{1}{\sqrt{x^2 - 9}} \) to isolate \( \frac{dy}{dx} \).
Express the differential equation as \( dy = \frac{1}{\sqrt{x^2 - 9}} dx \) to prepare for integration on both sides.
Integrate both sides: \( \int dy = \int \frac{1}{\sqrt{x^2 - 9}} dx \). The left side integrates to \( y \), and the right side requires recognizing the integral form.
Recall that \( \int \frac{1}{\sqrt{x^2 - a^2}} dx = \ln|x + \sqrt{x^2 - a^2}| + C \). Use this formula with \( a = 3 \) to write the general solution for \( y \).
Apply the initial condition \( y(5) = \ln 3 \) by substituting \( x = 5 \) and \( y = \ln 3 \) into the general solution to solve for the constant \( C \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Separable Differential Equations
A separable differential equation can be written as a product of a function of x and a function of y, allowing variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.
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Solving Separable Differential Equations
Initial Value Problems (IVP)
An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition allows determination of the unique solution curve passing through the given point.
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Integration Techniques and Inverse Functions
Solving the differential equation involves integrating expressions that may include radicals like √(x² - 9). Recognizing standard integral forms and applying inverse hyperbolic or logarithmic functions is essential to express the solution explicitly.
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Related Practice
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 √(2x - x^2) dx
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (sec t + cot t)² dt
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx
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Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-y) cos(y) dy