Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1/(3³√x)
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 4.7.6b
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1 / 2x³
Guida verificata passo dopo passo1
Identify the function to find the antiderivative of, which is \(\frac{1}{2}x^{3}\).
Recall the power rule for antiderivatives: for \(x^{n}\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).
Apply the power rule to \(x^{3}\): increase the exponent by 1 to get 4, then divide by the new exponent, so the antiderivative of \(x^{3}\) is \(\frac{x^{4}}{4} + C\).
Since the original function is multiplied by \(\frac{1}{2}\), multiply the antiderivative by \(\frac{1}{2}\) to get \(\frac{1}{2} \cdot \frac{x^{4}}{4} + C\).
Simplify the expression to write the antiderivative as \(\frac{x^{4}}{8} + C\). Remember to verify your answer by differentiating it to check if you get back the original function.
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Antiderivative (Indefinite Integral)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed with an arbitrary constant since differentiation of a constant is zero.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for finding antiderivatives of polynomial functions like x³.
Video consigliato:
Percorso guidato
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps avoid mistakes in integration.
Video consigliato:
Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫3(2x + 1)² dx = (2x + 1)³ + C
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Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
4 sec 3x tan 3x
32
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Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/2)x⁻³ᐟ²
19
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Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = x(x − 1)
194
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Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = 1− 4/x², x ≠ 0
161
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