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
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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 4.7.9c
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
Guida verificata passo dopo passo1
Identify the function to find the antiderivative of: \(-\frac{1}{3} x^{-\frac{4}{3}}\).
Recall the power rule for antiderivatives: For \(f(x) = x^n\), an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule by adding 1 to the exponent: \(-\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}\).
Divide the original coefficient by the new exponent: Multiply \(-\frac{1}{3}\) by \(\frac{1}{-\frac{1}{3}}\) which is the reciprocal of the new exponent.
Write the antiderivative as \(F(x) = -\frac{1}{3} \times \frac{x^{-\frac{1}{3}}}{-\frac{1}{3}} + C\), then simplify the expression and add the constant of integration \(C\).
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Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using basic integration rules. The result includes a constant of integration 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 states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating functions with variable exponents, including negative and fractional powers, by increasing the exponent by one and dividing by the new exponent.
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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 identify any mistakes in the integration process.
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Percorso guidato
05:53Finding Differentials
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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.
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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?
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Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
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Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
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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.
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