Skip to main content
Ch. 4 - Applications of the Derivative
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. 4 - Applications of the DerivativeProblema 4.9.55
Capitolo 4, Problema 4.9.55

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (4/x√(x² - 1))dx

Guida verificata passo dopo passo
1
Rewrite the integral in a more convenient form: \( \int \frac{4}{x\sqrt{x^2 - 1}} \, dx \). Notice that the integrand suggests a substitution involving a trigonometric identity.
Use the substitution \( x = \sec(\theta) \), which implies \( dx = \sec(\theta)\tan(\theta) \, d\theta \) and \( \sqrt{x^2 - 1} = \tan(\theta) \). Substitute these into the integral.
After substitution, the integral becomes \( \int \frac{4}{\sec(\theta) \cdot \tan(\theta)} \cdot \sec(\theta)\tan(\theta) \, d\theta \). Simplify the expression by canceling terms.
The simplified integral is \( \int 4 \, d\theta \), which is straightforward to integrate. Perform the integration to get \( 4\theta + C \), where \( C \) is the constant of integration.
Finally, reverse the substitution \( x = \sec(\theta) \) to express \( \theta \) in terms of \( x \). Since \( \theta = \sec^{-1}(x) \), the result becomes \( 4\sec^{-1}(x) + C \). Verify the solution by differentiating it to ensure it matches the original integrand.

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:
3m

Concetti chiave

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Integration Techniques

Various techniques are used to solve integrals, including substitution, integration by parts, and trigonometric identities. For the integral ∫ (4/x√(x² - 1))dx, recognizing the form of the integrand can suggest a suitable method, such as trigonometric substitution, to simplify the integration process.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the obtained function. This process ensures that the derivative of the integrated function returns to the original integrand, confirming the correctness of the integration. This step is crucial for validating the solution in calculus.
Video consigliato:
Percorso guidato
05:53
Finding Differentials
Pratica correlata
Domanda del libro di testo

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = sin 3x on [-π/4,π/3]

204
views
Domanda del libro di testo

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

183
views
Domanda del libro di testo

Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.


a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10

94
views
Domanda del libro di testo

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.


f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)

302
views
Domanda del libro di testo

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = √2 sin x- x on [0, 2π]

253
views
Domanda del libro di testo

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = 4√x and y = x² + 1

242
views