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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.16b
Capitolo 4, Problema 4.7.16b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
4 sec 3x tan 3x

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Identify the function to find the antiderivative of: \(\sec^3(3x) \tan(3x)\).
Recall that the derivative of \(\sec(x)\) is \(\sec(x) \tan(x)\), and the derivative of \(\sec^n(x)\) involves powers of \(\sec(x)\) and \(\tan(x)\).
Use substitution: let \(u = 3x\), so \(du = 3 \, dx\) or \(dx = \frac{du}{3}\).
Rewrite the integral in terms of \(u\): \(\int \sec^3(u) \tan(u) \cdot \frac{du}{3}\).
Recognize that \(\frac{1}{3}\) is a constant factor and the integral becomes \(\frac{1}{3} \int \sec^3(u) \tan(u) \, du\). Then, use the substitution \(w = \sec(u)\), so \(dw = \sec(u) \tan(u) \, du\), to express the integral in terms of \(w\) and integrate.

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Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as indefinite integrals with a constant of integration. This process helps solve problems involving accumulation and area under curves.
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Introduction to Indefinite Integrals

Trigonometric Functions and Their Derivatives

Understanding the derivatives of trigonometric functions like secant and tangent is essential. For example, the derivative of tan(x) is sec²(x), and the derivative of sec(x) is sec(x)tan(x). Recognizing these relationships aids in finding antiderivatives involving products of trig functions.
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Derivatives of Other Inverse Trigonometric Functions

Integration Techniques: Substitution and Product Rule Reversal

When integrating products of functions, techniques like substitution or recognizing derivatives of products (reverse product rule) are useful. For instance, if the integrand resembles the derivative of a product of functions, identifying this can simplify finding the antiderivative.
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The Product Rule
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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/(3³√x)

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Domanda del libro di testo

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 / 2x³

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Domanda del libro di testo

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫3(2x + 1)² dx = (2x + 1)³ + C

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Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


b. Show that the only local extreme value of f occurs at x = 2.

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Domanda del libro di testo

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻³/2 + x²

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Domanda del libro di testo

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)²(x + 2)

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