Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.82
Capitolo 5, Problema 5.5.82

Variations on the substitution method Evaluate the following integrals.                                                                                                        
                                                                                                                                                                    
 ∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

Guida verificata passo dopo passo
1
Recognize that the integral is of the form \(\int \frac{e^x - e^{-x}}{e^x + e^{-x}} \, d\!x\), which suggests a substitution involving the denominator or a related function.
Recall the hyperbolic functions: \(\sinh x = \frac{e^x - e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). Notice that the numerator is \(2 \sinh x\) and the denominator is \(2 \cosh x\), so the integrand simplifies to \(\frac{2 \sinh x}{2 \cosh x} = \frac{\sinh x}{\cosh x} = \tanh x\).
Rewrite the integral as \(\int \tanh x \, d\!x\) to simplify the problem.
Recall that the derivative of \(\ln(\cosh x)\) is \(\tanh x\), so the integral of \(\tanh x\) with respect to \(x\) is \(\ln|\cosh x| + C\).
Write the final integral expression as \(\int \frac{e^x - e^{-x}}{e^x + e^{-x}} \, d\!x = \ln|\cosh x| + C\), where \(C\) is the constant of integration.

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:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform a complicated integral into a basic form. It involves identifying a part of the integrand as a new variable, differentiating it, and rewriting the integral in terms of this variable. This technique is especially useful when the integral contains composite functions.
Video consigliato:
07:33
Euler's Method

Hyperbolic Functions and Their Properties

Expressions involving eˣ and e⁻ˣ often relate to hyperbolic functions such as sinh(x) and cosh(x). Recognizing these can simplify integration since sinh(x) = (eˣ - e⁻ˣ)/2 and cosh(x) = (eˣ + e⁻ˣ)/2. Using these identities helps rewrite the integral in a more manageable form.
Video consigliato:
Percorso guidato
06:21
Properties of Functions

Integration of Rational Functions

Integrals involving ratios of functions, like (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ), require understanding how to manipulate and simplify rational expressions. This often involves algebraic simplification or substitution to reduce the integral to a standard form, enabling straightforward integration.
Video consigliato:
6:04
Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.


The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π

103
views
Domanda del libro di testo

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.

v = [1 / (2t + 1)] (m/s), for 0 ≤ t ≤ 8 ; n = 4

52
views
Domanda del libro di testo

Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

123
views
Domanda del libro di testo

Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2                                                                                                                                                                       

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(f) Find a constant C such that F(𝓍) = G(𝓍) + C .

48
views
Domanda del libro di testo

Evaluating integrals Evaluate the following integrals.


∫₀⁵ |2𝓍―8|d𝓍

49
views
Domanda del libro di testo

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.


 ∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5      if 𝓍 ≤ 2                                                                                                                                                                                     

                      3𝓍 ― 1  if 𝓍 > 2

147
views