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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.66a
Capitolo 4, Problema 4.7.66a

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)

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1
Identify the integral to be checked: \(\int \sqrt{2x + 1} \, dx\) and the proposed antiderivative \(\sqrt{x^2 + x + C}\).
Recall that to verify if a function \(F(x)\) is an antiderivative of \(f(x)\), we differentiate \(F(x)\) and check if \(F'(x) = f(x)\).
Differentiate the proposed antiderivative \(F(x) = \sqrt{x^2 + x + C}\) using the chain rule: \(F'(x) = \frac{1}{2\sqrt{x^2 + x + C}} \cdot (2x + 1)\).
Compare \(F'(x)\) with the original integrand \(\sqrt{2x + 1}\). Notice that \(F'(x)\) has a denominator involving \(\sqrt{x^2 + x + C}\), which is not present in the original integrand.
Conclude that since \(F'(x) \neq \sqrt{2x + 1}\), the proposed antiderivative is incorrect.

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Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). The indefinite integral symbol ∫ represents the family of all antiderivatives, including a constant of integration C, since differentiation of a constant is zero.
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Introduction to Indefinite Integrals

Integration of Composite Functions and Substitution

When integrating functions like √(2x + 1), substitution is often used by setting u = 2x + 1 to simplify the integral. This method helps handle composite functions by changing variables, making the integral easier to evaluate correctly.
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Indefinite Integrals

Verification of Antiderivative by Differentiation

To check if a proposed antiderivative is correct, differentiate it and see if the result matches the original integrand. If the derivative does not equal the integrand, the formula is incorrect.
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Antiderivatives
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Solve the initial value problems in Exercises 89–92.


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Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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