Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx
31
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 8, Problema 8.4.63
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
Guida verificata passo dopo passo1
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.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
Percorso guidato
06:37
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.
Video consigliato:
Percorso guidato
05:43Definition of the Definite Integral
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.
Video consigliato:
06:13Limits of Rational Functions with Radicals
Pratica correlata
Domanda del libro di testo
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.
∫ dx / (x - √x)
30
views
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to π/2 of (cot θ dθ)
5
views
Domanda del libro di testo
Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.
44
views
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 2 to ∞ of (dx / √(x² - 1))
29
views
Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)
38
views