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.
∫(3t² + t/2) dt
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 4.7.67
Checking Antiderivative Formulas
Right, or wrong? Give a brief reason why.
∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C
Verified step by step guidance1
Identify the given integral: \(\int -15 \frac{(x + 3)^2}{(x - 2)^4} \, dx\) and the proposed antiderivative: \(\left( \frac{x + 3}{x - 2} \right)^3 + C\).
To verify if the proposed antiderivative is correct, differentiate \(\left( \frac{x + 3}{x - 2} \right)^3\) using the chain rule.
Let \(u = \frac{x + 3}{x - 2}\). Then the derivative of \(u^3\) with respect to \(x\) is \(3u^2 \cdot \frac{du}{dx}\).
Find \(\frac{du}{dx}\) by applying the quotient rule: \(\frac{d}{dx} \left( \frac{x + 3}{x - 2} \right) = \frac{(1)(x - 2) - (x + 3)(1)}{(x - 2)^2} = \frac{x - 2 - x - 3}{(x - 2)^2} = \frac{-5}{(x - 2)^2}\).
Combine these results: \(\frac{d}{dx} \left( \frac{x + 3}{x - 2} \right)^3 = 3 \left( \frac{x + 3}{x - 2} \right)^2 \cdot \left( \frac{-5}{(x - 2)^2} \right) = -15 \frac{(x + 3)^2}{(x - 2)^4}\), which matches the integrand, confirming the antiderivative is correct.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative and Indefinite Integration
An antiderivative of a function is another function whose derivative equals the original function. Indefinite integration finds all antiderivatives and includes a constant of integration (C) because differentiation loses constant terms. Verifying an antiderivative involves differentiating the proposed solution and checking if it matches the original integrand.
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Introduction to Indefinite Integrals
Chain Rule and Power Rule in Differentiation
The chain rule is used to differentiate composite functions, applying the derivative of the outer function multiplied by the derivative of the inner function. The power rule states that d/dx[x^n] = n*x^(n-1). Understanding these rules is essential to verify if the given antiderivative is correct by differentiating it properly.
Recommended video:
05:02Intro to the Chain Rule
Algebraic Manipulation of Rational Functions
Rational functions are ratios of polynomials, and simplifying or rewriting them can help in integration or differentiation. Recognizing how to express the integrand and the proposed antiderivative in comparable forms is crucial to verify correctness, especially when powers and sums are involved in numerator and denominator.
Recommended video:
06:11Limits of Rational Functions: Denominator = 0
Related Practice
Textbook Question
The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?
Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = x¹ᐟ³(x + 8)
Textbook Question
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
__________
y = √ 3 + 2𝓍 ―𝓍²
Textbook Question
Initial Value Problems
Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 1/x² + x, x > 0; y(2) = 1