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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 4.7.31
Chapter 4, Problem 4.7.31

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.


∫2x(1 − x⁻³) dx

Verified step by step guidance
1
First, rewrite the integrand to simplify the expression. Distribute \$2x$ across the terms inside the parentheses: \(2x(1 - x^{-3}) = 2x - 2x \cdot x^{-3}\).
Simplify the second term by combining the powers of \(x\): \(2x - 2x^{1 + (-3)} = 2x - 2x^{-2}\).
Express the integral as the sum of two separate integrals: \(\int 2x \, dx - \int 2x^{-2} \, dx\).
Integrate each term using the power rule for integration, which states \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\).
After integrating, combine the results and add the constant of integration \(C\). Remember to verify your answer by differentiating it to ensure it matches the original integrand.

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Key Concepts

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

Indefinite Integral (Antiderivative)

An indefinite integral represents the most general form of the antiderivative of a function, including a constant of integration. It reverses differentiation and is expressed without limits, showing all possible functions whose derivative matches the integrand.
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Introduction to Indefinite Integrals

Integration by Simplification

This technique involves algebraically manipulating the integrand to a simpler form before integrating. For example, expanding or rewriting expressions like 2x(1 − x⁻³) into separate terms makes it easier to apply basic integration rules.
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Integration by Parts for Definite Integrals

Verification by Differentiation

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


∫(sin2x − csc²x)dx

Textbook Question

Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {sinx / x, −π ≤ x < 0

0, x = 0

Textbook Question

Finding Extrema from Graphs


In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.


f(x) = |x|, −1 < x < 2

Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.


f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8

Textbook Question

Absolute Extrema on Finite Closed Intervals


In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.


h(x) = ³√x, −1 ≤ x ≤ 8