Domanda del libro di testo
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ₑᵉ^³ dx / (x ln x ln²(ln x))
49
views
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testo
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.63
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.63Capitolo 7, Problema 7.3.63
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁ᵉ^² dx/x√(ln²x + 1)
Guida verificata passo dopo passo1
Identify the integral to be evaluated: \(\displaystyle \int_1^{e^2} \frac{1}{x \sqrt{(\ln x)^2 + 1}} \, dx\).
Recognize that the integrand involves \(\ln x\) and its derivative. Consider the substitution \(t = \ln x\), which implies \(dt = \frac{1}{x} dx\).
Rewrite the integral in terms of \(t\): since \(x\) goes from 1 to \(e^2\), then \(t\) goes from \(\ln 1 = 0\) to \(\ln e^2 = 2\). The integral becomes \(\int_0^2 \frac{1}{\sqrt{t^2 + 1}} \, dt\).
Recall that \(\int \frac{1}{\sqrt{t^2 + 1}} \, dt\) is a standard integral whose antiderivative is \(\sinh^{-1}(t)\) or equivalently \(\ln(t + \sqrt{t^2 + 1})\) (this is Theorem 7.7, expressing inverse hyperbolic sine in terms of logarithms).
Evaluate the definite integral by substituting the limits \(t=0\) and \(t=2\) into the antiderivative \(\ln(t + \sqrt{t^2 + 1})\) and subtracting: \(\left. \ln(t + \sqrt{t^2 + 1}) \right|_0^2\).
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. Typically, a substitution u = g(x) is chosen so that du replaces part of the integrand, making integration straightforward. This method is especially useful when the integrand contains composite functions.
Video consigliato:
07:33Euler's Method
Logarithmic Functions and Their Properties
Logarithmic functions, such as ln(x), are inverses of exponential functions and have properties that simplify expressions involving products, quotients, and powers. In integration, expressing answers in terms of logarithms often involves recognizing derivatives of ln(x) and using log rules to rewrite results clearly.
Video consigliato:
Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
7–28. Derivatives Evaluate the following derivatives.
d/dx (e^{-10x²})
59
views
Domanda del libro di testo
101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y
75
views
Domanda del libro di testo
7–28. Derivatives Evaluate the following derivatives.
d/dy (y^{sin y})
36
views
Domanda del libro di testo
11–15. Identities Prove each identity using the definitions of the hyperbolic functions.
cosh 2x = cosh²x + sinh²x (Hint: Begin with the right side of the equation.)
122
views
Domanda del libro di testo
Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.
91
views