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.
-(1/3)x⁻⁴ᐟ³
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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 4.7.1c
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x² − 2x + 1
Guida verificata passo dopo passo1
Identify the function to find the antiderivative of: \(x^{2} - 2x + 1\).
Recall that the antiderivative (indefinite integral) of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). We will integrate each term separately.
Use the power rule for integration: For any term \(x^{n}\), the antiderivative is \(\frac{x^{n+1}}{n+1}\), provided \(n \neq -1\).
Integrate each term: \(\int x^{2} \, dx = \frac{x^{3}}{3}\), \(\int (-2x) \, dx = -2 \cdot \frac{x^{2}}{2} = -x^{2}\), and \(\int 1 \, dx = x\).
Combine the results and add the constant of integration \(C\): \(F(x) = \frac{x^{3}}{3} - x^{2} + x + C\). This is the general antiderivative of the given 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 often expressed with an arbitrary constant, C, 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) + C. This rule is essential for integrating polynomial terms like x² and x.
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.
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Percorso guidato
05:53Finding Differentials
Pratica correlata
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.
√x + 1/√x
29
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
137
views
Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = (1/2) sec²θ + C
26
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)²(x + 2)
157
views
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.
−π csc (πx/2) cot (πx/2)
30
views