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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.16b
Chapter 4, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

Identifying Extrema


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


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b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Textbook Question

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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Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Textbook Question

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

Textbook Question

Finding Antiderivatives

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

x⁷

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