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
1
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 4.7.5a
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 \(\frac{1}{x^2}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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. 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.
Recommended video:
05:04
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).
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.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫√(2x + 1) dx = √(x² + x +C)
1
views
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) exist?
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
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
1
views