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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 4.7.19
Chapter 4, Problem 4.7.19

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.


∫(3t² + t/2) dt

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Identify the integral to solve: \(\int \left(3t^{2} + \frac{t}{2}\right) \, dt\).
Split the integral into the sum of two separate integrals: \(\int 3t^{2} \, dt + \int \frac{t}{2} \, dt\).
Apply the power rule for integration to each term separately. Recall that \(\int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C\) for \(n \neq -1\).
Integrate the first term: \(\int 3t^{2} \, dt = 3 \int t^{2} \, dt = 3 \cdot \frac{t^{3}}{3} = t^{3}\).
Integrate the second term: \(\int \frac{t}{2} \, dt = \frac{1}{2} \int t \, dt = \frac{1}{2} \cdot \frac{t^{2}}{2} = \frac{t^{2}}{4}\), then combine the results and add 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 family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C. It does not have specified limits, unlike definite integrals.
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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 is fundamental for integrating polynomial terms like t² or t/2 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

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

39. y = 8 / (x² + 4) (Witch of Agnesi)

Textbook Question

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x¹ᐟ³(x + 8)

Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.


y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Textbook Question

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Give a brief reason why.


∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C

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