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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 4.7.43
Chapter 4, Problem 4.7.43

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.


∫(4secx tanx − 2 sec²x)dx

Verified step by step guidance
1
Recognize that the integral is of the form \(\int (4 \sec x \tan x - 2 \sec^{2} x) \, dx\), which can be split into two separate integrals: \(\int 4 \sec x \tan x \, dx - \int 2 \sec^{2} x \, dx\).
Recall the standard derivatives: \(\frac{d}{dx}(\sec x) = \sec x \tan x\) and \(\frac{d}{dx}(\tan x) = \sec^{2} x\). This helps identify antiderivatives for each term.
For the first integral, \(\int 4 \sec x \tan x \, dx\), use the fact that the derivative of \(\sec x\) is \(\sec x \tan x\), so the antiderivative is \(4 \sec x\) plus a constant.
For the second integral, \(\int 2 \sec^{2} x \, dx\), use the fact that the derivative of \(\tan x\) is \(\sec^{2} x\), so the antiderivative is \(2 \tan x\) plus a constant.
Combine the results from both integrals and add a single constant of integration \(C\) to write the most general antiderivative: \(4 \sec x - 2 \tan x + C\).

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Key Concepts

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

Indefinite Integral and Antiderivative

An indefinite integral represents the most general form of an antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.
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Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like secant and tangent requires knowledge of their derivatives and standard integral formulas. Recognizing patterns such as the derivative of sec x being sec x tan x aids in simplifying and solving the integral efficiently.
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Introduction to Trigonometric Functions

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the solution and helps identify any errors in the integration process, ensuring the integral is accurate.
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Related Practice
Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

8. y = 2cosx - √2x, -π≤x≤3π/2

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

85. y' = x^(-2/3) (x - 1)

Textbook Question

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.


∫(1/x² − x² − 1/3) dx

Textbook Question

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).

Textbook Question

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

Textbook Question

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.


∫(−3csc²x)dx