Domanda del libro di testo
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
__________
y = √ 3 + 2𝓍 ―𝓍²
226
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.7.27
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(√x + ³√x) dx
Guida verificata passo dopo passo1
Rewrite the integral in terms of exponents to make it easier to integrate. Recall that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). So the integral becomes \( \int \left( x^{1/2} + x^{1/3} \right) \, dx \).
Use the power rule for integration, which states that for any real number \( n \neq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Apply the power rule separately to each term inside the integral: \( \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C_1 = \frac{x^{3/2}}{3/2} + C_1 \) and \( \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C_2 = \frac{x^{4/3}}{4/3} + C_2 \).
Simplify the fractions in the denominators by multiplying numerator and denominator appropriately: \( \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \) and \( \frac{x^{4/3}}{4/3} = \frac{3}{4} x^{4/3} \).
Combine the results and write the most general antiderivative as \( \frac{2}{3} x^{3/2} + \frac{3}{4} x^{4/3} + C \), where \( C \) is the constant of integration. To verify, differentiate this expression and check if you get back 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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Indefinite Integral (Antiderivative)
An indefinite integral represents the most general form of the antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(x) dx = F(x) + C, where F'(x) = f(x).
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like √x (x^(1/2)) and ³√x (x^(1/3)) by rewriting roots as fractional exponents and applying the formula.
Video consigliato:
Percorso guidato
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution. It ensures no algebraic errors occurred during integration and that the constant of integration is properly accounted for.
Video consigliato:
Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)
212
views
Domanda del libro di testo
6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0,b). Show that the area of the triangle enclosed by the segment is largest when a = b.
213
views
Domanda del libro di testo
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²y/dx² = 2 − 6x; y′(0) = 4, y(0) = 1
23
views
Domanda del libro di testo
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
g(x) = x√8 − x²
135
views
Domanda del libro di testo
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(t√t + √t) / t² dt
30
views