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 4.9.29
Capitolo 4, Problema 4.9.29

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


∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

Guida verificata passo dopo passo
1
Step 1: Recall the power rule for integration, which states that for any term of the form ∫xⁿ dx, the integral is (xⁿ⁺¹)/(n+1) + C, where n ≠ -1. Apply this rule to each term in the integrand.
Step 2: For the first term, 3x¹⸍³, increase the exponent by 1 (1/3 + 1 = 4/3) and divide by the new exponent. The integral becomes (3x⁴⸍³)/(4/3). Simplify this expression.
Step 3: For the second term, 4x⁻¹⸍³, increase the exponent by 1 (-1/3 + 1 = 2/3) and divide by the new exponent. The integral becomes (4x²⸍³)/(2/3). Simplify this expression.
Step 4: For the constant term, 6, recall that the integral of a constant is the constant multiplied by x. Thus, the integral of 6 is 6x.
Step 5: Combine all the results from the previous steps and add the constant of integration, C, to form the final indefinite integral. Check your work by differentiating the result 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 often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Power Rule for Integration

The power rule for integration is a fundamental technique used to integrate polynomial functions. It states that for any real number n ≠ -1, the integral of x^n with respect to x is (x^(n+1))/(n+1) + C. This rule simplifies the process of integrating terms like 3x^(1/3) and 4x^(-1/3) in the given integral.
Video consigliato:
Percorso guidato
04:04
Power Rule for Indefinite Integrals

Checking Work by Differentiation

To verify the correctness of an indefinite integral, one can differentiate the result obtained from the integration. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This process reinforces the relationship between differentiation and integration, as they are inverse operations.
Video consigliato:
Percorso guidato
06:22
Introduction To Work
Pratica correlata
Domanda del libro di testo

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = tan x

242
views
Domanda del libro di testo

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = e⁻ˣ - ((x + 4)/5)

208
views
Domanda del libro di testo

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos 2x - x² + 2x

228
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) = cos² x on [-π,π]

227
views
Domanda del libro di testo

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - 1) / (2x + 5)

241
views
Domanda del libro di testo

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→∞ (e¹/ₓ - 1)/(1/x)

201
views