Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
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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 8.4.63
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
Verified step by step guidance1
Recall the formula for the average value of a function \(f(x)\) on the interval \([a, b]\):
\[\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx\]
Identify the function and interval: here, \(f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}}\) and the interval is \([1, 3]\). So, \(a = 1\) and \(b = 3\).
Rewrite the function to simplify the integral:
\[f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}} = \sqrt{\frac{x + 1}{x}} = \sqrt{1 + \frac{1}{x}}\]
Set up the integral for the average value:
\[\frac{1}{3 - 1} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx = \frac{1}{2} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx\]
To evaluate the integral, consider substitution or algebraic manipulation to simplify the integrand before integrating. After finding the integral, multiply by \(\frac{1}{2}\) to get the average value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Value of a Function
The average value of a function f(x) on an interval [a, b] is given by (1/(b - a)) times the definite integral of f(x) from a to b. It represents the mean height of the function over that interval.
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Average Value of a Function
Definite Integral
A definite integral calculates the net area under the curve of a function between two points a and b. It is essential for finding the total accumulation or average value of a function over an interval.
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Simplifying and Integrating Radical Functions
Functions involving square roots often require algebraic manipulation to simplify before integration. Techniques include rewriting radicals as fractional exponents and applying integration rules for power functions.
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Related Practice
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
Textbook Question
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = c * x * √(25 - x²) over [0, 5]
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Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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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.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).