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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.77
Chapter 4, Problem 4.PE.77

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

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Recognize that the integral is of the form \(\int \frac{1}{(r + 5)^2} \, dr\), which can be rewritten as \(\int (r + 5)^{-2} \, dr\) to make it easier to apply the power rule for integration.
Recall the power rule for integration: for any real number \(n \neq -1\), \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). Here, \(n = -2\).
Apply the power rule to the integral: \(\int (r + 5)^{-2} \, dr = \frac{(r + 5)^{-2 + 1}}{-2 + 1} + C = \frac{(r + 5)^{-1}}{-1} + C\).
Since the integral involves a composite function \((r + 5)\), use substitution to account for the inner function's derivative. Let \(u = r + 5\), then \(du = dr\). This confirms the integral with respect to \(u\) is straightforward.
Write the final antiderivative as \(- (r + 5)^{-1} + C\), or equivalently \(- \frac{1}{r + 5} + C\). Remember to include the constant of integration \(C\).

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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 with a constant of integration (C). It reverses differentiation and provides a family of functions whose derivative is the integrand.
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Introduction to Indefinite Integrals

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 helps integrate functions with variables raised to a power, including negative exponents.
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Power Rule for Indefinite Integrals

Chain Rule and Substitution Method

When integrating composite functions like 1/(r+5)^2, substitution simplifies the integral by setting u = r + 5. This method reverses the chain rule from differentiation, making integration manageable.
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Intro to the Chain Rule
Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -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.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

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.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = (x + 1)², −∞ < x ≤ 0

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.

__

∫ ( 3√ t + 4/t² ) dt

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