Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π sin⁵(x/2) dx
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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 8.5.70
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0
Verified step by step guidance1
Rewrite the given differential equation \((t + 1) \frac{dx}{dt} = x^{2} + 1\) to isolate \(\frac{dx}{dt}\):
\(\frac{dx}{dt} = \frac{x^{2} + 1}{t + 1}\).
Recognize that this is a separable differential equation. Rearrange terms to separate variables \(x\) and \(t\):
\(\frac{dx}{x^{2} + 1} = \frac{dt}{t + 1}\).
Integrate both sides:
\(\int \frac{dx}{x^{2} + 1} = \int \frac{dt}{t + 1}\).
Evaluate the integrals:
The left integral is \(\arctan(x)\), and the right integral is \(\ln|t + 1| + C\), where \(C\) is the constant of integration. So,
\(\arctan(x) = \ln|t + 1| + C\).
Use the initial condition \(x(0) = 0\) to find \(C\):
Substitute \(t=0\) and \(x=0\) into the equation to solve for \(C\), then express \(x\) explicitly as a function of \(t\) by taking the tangent of both sides.
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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 so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side. This allows integration of both sides separately to find the solution. Recognizing and rearranging the given equation into separable form is essential for solving it.
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Solving Separable Differential Equations
Initial Value Problems (IVP)
An initial value problem specifies the value of the unknown function at a particular point, which helps determine the unique solution to a differential equation. Using the initial condition x(0) = 0 allows us to find the constant of integration after solving the differential equation.
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05:03Initial Value Problems
Integration Techniques
Solving the separated differential equation requires integrating expressions involving variables and constants. Familiarity with standard integration methods, such as partial fractions or substitution, is necessary to evaluate the integrals and express x as a function of t.
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06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (x² - 2x + 1) e^(2x) dx
Textbook Question
In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy
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Textbook Question
Volume of water in a swimming pool
A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral
V = ∫ from 0 to 50 of 30 · h(x) dx.
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² tan⁻¹(x / 2) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)
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