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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 4.7.23
Chapter 4, Problem 4.7.23

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

Verified step by step guidance
1
Rewrite the integral by expressing each term with exponents: \(\int \left( x^{-2} - x^{2} - \frac{1}{3} \right) \, dx\).
Use the linearity of the integral to split it into separate integrals: \(\int x^{-2} \, dx - \int x^{2} \, dx - \int \frac{1}{3} \, dx\).
Apply the power rule for integration to each term: For \(\int x^{n} \, dx\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Integrate each term individually: \(\int x^{-2} \, dx = \frac{x^{-1}}{-1} + C\), \(\int x^{2} \, dx = \frac{x^{3}}{3} + C\), and \(\int \frac{1}{3} \, dx = \frac{1}{3}x + C\).
Combine the results, remembering to include a single constant of integration \(+ C\) at the end, and write the most general antiderivative.

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Key Concepts

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

Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation and is written as ∫f(x) dx, yielding a function F(x) such that F'(x) = f(x).
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Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating polynomial terms by increasing the exponent by one and dividing by the new exponent.
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Power Rule for Indefinite Integrals

Verification by Differentiation

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

35. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

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.


∫(4secx tanx − 2 sec²x)dx

Textbook Question

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.

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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