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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 4.7.58
Capitolo 4, Problema 4.7.58

Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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Identify the given antiderivative formula: \(\int (3x + 5)^{-2} \, dx = -\frac{(3x + 5)^{-1}}{3} + C\).
To verify this formula, differentiate the right-hand side expression \(-\frac{(3x + 5)^{-1}}{3} + C\) with respect to \(x\).
Apply the chain rule for differentiation: if \(f(x) = (3x + 5)^{-1}\), then \(f'(x) = -1 \cdot (3x + 5)^{-2} \cdot 3\) because the derivative of the inside function \(3x + 5\) is 3.
Multiply the derivative by the constant factor \(-\frac{1}{3}\) outside the function, simplifying the expression step-by-step.
Confirm that the resulting derivative simplifies exactly to the original integrand \((3x + 5)^{-2}\), which verifies the antiderivative formula.

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Antiderivatives and Indefinite Integrals

An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C, since differentiation of a constant is zero.
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Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule helps integrate functions with variables raised to a power, including negative exponents, by increasing the exponent by one and dividing by the new exponent.
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Verification by Differentiation

To verify an antiderivative formula, differentiate the proposed antiderivative and check if it equals the original integrand. This process confirms the correctness of the integral by applying the fundamental theorem of calculus in reverse.
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Finding Differentials
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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.


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Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


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Roots (Zeros)


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Finding Indefinite Integrals


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