Domanda del libro di testo
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
71
views
Ch. 4 - Applications of the Derivative
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
Capitolo 4, Problema 4.R.99
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (12/x)dx
Guida verificata passo dopo passo1
Step 1: Recognize that the integral ∫ (12/x) dx involves a basic logarithmic rule. Recall that the integral of 1/x with respect to x is ln|x|.
Step 2: Factor out the constant 12 from the integral. This simplifies the expression to 12 ∫ (1/x) dx.
Step 3: Apply the logarithmic integration rule. The integral of 1/x dx is ln|x|, so the expression becomes 12 ln|x|.
Step 4: Add the constant of integration, C, to account for the indefinite nature of the integral. The result is 12 ln|x| + C.
Step 5: Verify the result by differentiating 12 ln|x| + C. The derivative of ln|x| is 1/x, and multiplying by 12 gives back the original integrand, confirming the solution.
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:
56sConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Indefinite Integral
An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without specific limits and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration of Rational Functions
Rational functions are ratios of polynomials. To integrate a rational function, one often uses techniques such as substitution or partial fraction decomposition. In the case of the integral ∫ (12/x)dx, recognizing that this is a simple rational function allows for straightforward integration using the natural logarithm.
Video consigliato:
6:04Intro to Rational Functions
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately 2.71828. It is particularly important in calculus because the derivative of ln(x) is 1/x, making it a key function when integrating expressions involving 1/x, such as in the integral ∫ (12/x)dx.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).
254
views
Domanda del libro di testo
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
195
views
Domanda del libro di testo
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
79
views
Domanda del libro di testo
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 2Θ cot 3Θ
210
views
Domanda del libro di testo
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 10x² / (x² + 3)
214
views