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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 4.7.27
Chapter 4, Problem 4.7.27

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.


∫(√x + ³√x) dx

Verified step by step guidance
1
Rewrite the integral in terms of exponents to make it easier to integrate. Recall that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). So the integral becomes \( \int \left( x^{1/2} + x^{1/3} \right) \, dx \).
Use the power rule for integration, which states that for any real number \( n \neq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Apply the power rule separately to each term inside the integral: \( \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C_1 = \frac{x^{3/2}}{3/2} + C_1 \) and \( \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C_2 = \frac{x^{4/3}}{4/3} + C_2 \).
Simplify the fractions in the denominators by multiplying numerator and denominator appropriately: \( \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \) and \( \frac{x^{4/3}}{4/3} = \frac{3}{4} x^{4/3} \).
Combine the results and write the most general antiderivative as \( \frac{2}{3} x^{3/2} + \frac{3}{4} x^{4/3} + C \), where \( C \) is the constant of integration. To verify, differentiate this expression and check if you get back 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, finding a function whose derivative matches the integrand. For example, ∫f(x) dx = F(x) + C, where F'(x) = f(x).
Recommended video:
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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 essential for integrating terms like √x (x^(1/2)) and ³√x (x^(1/3)) by rewriting roots as fractional exponents and applying the formula.
Recommended video:
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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. It ensures no algebraic errors occurred during integration and that the constant of integration is properly accounted for.
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Finding Differentials
Related Practice
Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

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.


∫(−2cost) dt

Textbook Question

Applications


Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?

Textbook Question

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


1
views
Textbook Question

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


g(x) = {x³, −2 ≤ x ≤ 0

x², 0 < x ≤ 2

Textbook Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?