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.
-(1/3)x⁻⁴ᐟ³
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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 4.7.7c
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
Verified step by step guidance1
Rewrite the given function in terms of exponents to make it easier to integrate. Recall that \(\sqrt{x} = x^{\frac{1}{2}}\) and \(\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}\). So the function becomes \(f(x) = x^{\frac{1}{2}} + x^{-\frac{1}{2}}\).
Recall the power rule for antiderivatives: for any real number \(n \neq -1\), the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to each term separately: for \(x^{\frac{1}{2}}\), add 1 to the exponent to get \(\frac{1}{2} + 1 = \frac{3}{2}\), then divide by \(\frac{3}{2}\). For \(x^{-\frac{1}{2}}\), add 1 to the exponent to get \(-\frac{1}{2} + 1 = \frac{1}{2}\), then divide by \(\frac{1}{2}\).
Write the antiderivative as the sum of the two results from the previous step, plus the constant of integration \(C\).
To check your answer, differentiate your antiderivative using the power rule for derivatives and verify that you get back the original function \(\sqrt{x} + \frac{1}{\sqrt{x}}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating functions involving powers of x, including fractional and negative exponents.
Recommended video:
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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05:53Finding Differentials
Related Practice
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.
x² − 2x + 1
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.
−π csc (πx/2) cot (πx/2)
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Textbook Question
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) = 1− 4/x², x ≠ 0
Textbook Question
[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).
27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.
c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)
Textbook Question
Dependence on Initial Point
8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations
c. x_0=2