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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 4.7.64c
Capitolo 4, Problema 4.7.64c

Checking Antiderivative Formulas


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


∫tanθ sec²θ dθ = (1/2) sec²θ + C

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1
Identify the integral to be evaluated: \(\int \tan\theta \sec^{2}\theta \, d\theta\).
Recall that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) and \(\sec\theta = \frac{1}{\cos\theta}\), so \(\sec^{2}\theta = \left(\frac{1}{\cos\theta}\right)^2 = \frac{1}{\cos^{2}\theta}\).
Consider using substitution: let \(u = \tan\theta\), then \(\frac{du}{d\theta} = \sec^{2}\theta\), which implies \(du = \sec^{2}\theta \, d\theta\).
Rewrite the integral in terms of \(u\): \(\int \tan\theta \sec^{2}\theta \, d\theta = \int u \, du\).
Integrate \(\int u \, du\) to get \(\frac{1}{2} u^{2} + C\), then substitute back \(u = \tan\theta\) to obtain \(\frac{1}{2} \tan^{2}\theta + C\). Compare this with the given formula \(\frac{1}{2} \sec^{2}\theta + C\) to determine if it is correct.

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a function whose derivative also appears in the integral, allowing the integral to be rewritten in a simpler form. This technique is essential for verifying antiderivative formulas involving composite functions.
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Substitution With an Extra Variable

Derivatives of Trigonometric Functions

Understanding the derivatives of trigonometric functions like tan(θ) and sec(θ) is crucial. For example, the derivative of tan(θ) is sec²(θ), and the derivative of sec(θ) is sec(θ)tan(θ). Recognizing these relationships helps in identifying correct antiderivatives and verifying integral formulas.
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Derivatives of Other Inverse Trigonometric Functions

Constant of Integration

When finding antiderivatives, it is important to include the constant of integration, denoted by C, because indefinite integrals represent a family of functions differing by a constant. This constant accounts for all possible vertical shifts of the antiderivative graph.
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Integration by Parts for Definite Integrals
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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.

-(1/3)x⁻⁴ᐟ³

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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.

x² − 2x + 1

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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⁻⁴ + 2x + 3

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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:


c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = x(x − 1)

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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:


c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = (x − 1)²(x + 2)

157
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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.

−π csc (πx/2) cot (πx/2)

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