Textbook Question
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
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.8b
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)
Verified step by step guidance1
Rewrite the given function in a form that is easier to integrate. Recall that the cube root of x is \(x^{1/3}\), so the function \(\frac{1}{3\sqrt[3]{x}}\) can be written as \(\frac{1}{3x^{1/3}}\).
Express the function as \(\frac{1}{3} x^{-1/3}\) by bringing the \(x^{1/3}\) in the denominator to the numerator with a negative exponent.
Use the power rule for antiderivatives, which states that 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 to \(\frac{1}{3} x^{-1/3}\) by increasing the exponent by 1: \(-\frac{1}{3} + 1 = \frac{2}{3}\), and then divide by the new exponent \(\frac{2}{3}\).
Multiply the constant \(\frac{1}{3}\) by the reciprocal of the new exponent \(\frac{3}{2}\) to find the coefficient of the antiderivative, and don't forget to add the constant of integration \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 derivatives of constants are zero.
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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions expressed as powers of x, including fractional and negative exponents.
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Simplifying Expressions with Radicals and Exponents
Radicals can be rewritten as fractional exponents to simplify integration. For example, the cube root of x is x^(1/3). Converting radicals to exponents allows the use of the power rule directly and makes mental calculation easier.
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