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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.PE.73
Chapter 4, Problem 4.PE.73

Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

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1
Identify the integral to solve: \(\int (x^{3} + 5x - 7) \, dx\).
Recall the power rule for integration: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
Apply the power rule to each term separately: integrate \(x^{3}\), \$5x$, and $-7$ individually.
For \(x^{3}\), the integral is \(\frac{x^{4}}{4}\); for \$5x$, treat the constant 5 as a multiplier and integrate $x$ to get \(\frac{x^{2}}{2}\), so the term becomes \(5 \times \frac{x^{2}}{2}\); for $-7$, integrate the constant to get $-7x$.
Combine all integrated terms and add the constant of integration \(C\) to 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 most general antiderivative of a function, expressed as a family of functions plus a constant of integration (C). It reverses differentiation and is written without limits, indicating all possible antiderivatives.
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Power Rule for Integration

The power rule states that the integral of x^n (where n β‰  -1) is (x^(n+1)) / (n+1) + C. This rule is essential for integrating polynomial terms like xΒ³ and x, allowing straightforward calculation of their antiderivatives.
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Verification by Differentiation

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

Initial Value Problems

Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)Β² , y(1)= 1

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sin 2x, 0 ≀ x ≀ Ο€

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 1/( r + 5)²dr

Textbook Question

Analyzing Functions from Derivatives


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


a. What are the critical points of f?


fβ€²(x) = x(x βˆ’ 1)

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ