Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
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 4.7.17
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(x + 1) dx
Verified step by step guidance1
Identify the integral to solve: \(\int (x + 1) \, dx\).
Recall the basic rule for integrating sums: \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\).
Split the integral into two separate integrals: \(\int x \, dx + \int 1 \, dx\).
Use the power rule for integration on the first term: \(\int x \, dx = \frac{x^{2}}{2} + C_1\), where \(C_1\) is a constant of integration.
Integrate the constant term: \(\int 1 \, dx = x + C_2\), where \(C_2\) is another constant of integration. Combine constants into a single constant \(C\) to write the general antiderivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integral
An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C.
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05:04
Introduction to Indefinite Integrals
Basic Integration Rules
Integration rules such as the power rule and linearity simplify finding antiderivatives. For example, ∫x dx = x²/2 + C and ∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx, allowing term-by-term integration.
Recommended video:
06:07Basic Rules for Definite Integrals
Verification by Differentiation
After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and ensures no errors in the integration process.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫7sin(θ/3) dθ
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y = 1 / (x² - 1)
Textbook Question
Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.
c. f(x) = 1/√(1 + x)
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Textbook Question
Finding Functions from Derivatives
Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.