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.
−3x⁻⁴
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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 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
Chapter 4, Problem 4.7.5a
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 / x²
Verified step by step guidance1
Recognize that the function given is \(\frac{1}{x^2}\), which can be rewritten as \(x^{-2}\) to make it easier to apply the power rule for antiderivatives.
Recall the power rule for antiderivatives: for any function \(x^n\) where \(n \neq -1\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to \(x^{-2}\) by increasing the exponent by 1: \(-2 + 1 = -1\), so the antiderivative will be \(\frac{x^{-1}}{-1} + C\).
Simplify the expression to get the antiderivative in a more standard form: \(-x^{-1} + C\), which can also be written as \(-\frac{1}{x} + C\).
Verify your answer by differentiating \(-\frac{1}{x} + C\) and checking that the derivative is \(\frac{1}{x^2}\).
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2mKey Concepts
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Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed as an indefinite integral with a constant of integration, C, since differentiation loses constant terms.
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Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that for any real number n ≠ -1, the integral of x^n dx is (x^(n+1)) / (n+1) + C. This rule is essential for finding antiderivatives of polynomial and power functions like 1/x², which can be rewritten as x^(-2).
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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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Related Practice
Textbook Question
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = sec³θ / 3 + C
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Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = csc²x − 2cot x, 0 < x < π
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