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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.10
Chapter 5, Problem 5.5.10

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              
                                                                                                                                                                                        
 βˆ« (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

Verified step by step guidance
1
Step 1: Identify the substitution provided in the problem. Here, the substitution is u = 3𝓍² + 𝓍. Compute the derivative of u with respect to 𝓍: dπ“Š/d𝓍 = 6𝓍 + 1.
Step 2: Rewrite the integral using the substitution. Replace √(3𝓍² + 𝓍) with √u and (6𝓍 + 1)d𝓍 with dπ“Š, as dπ“Š = (6𝓍 + 1)d𝓍.
Step 3: The integral now becomes ∫ √u dπ“Š. This is a simpler integral to evaluate.
Step 4: Use the power rule for integration to solve ∫ √u dπ“Š. Recall that √u = u^(1/2), and the integral of u^(n) is (u^(n+1))/(n+1) + C, where C is the constant of integration.
Step 5: Substitute back u = 3𝓍² + 𝓍 into the result to express the solution in terms of 𝓍. Finally, check your answer by differentiating 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.

Substitution in Integration

Substitution is a technique used in integration to simplify the integral by changing the variable of integration. By letting u be a function of x, we can express the integral in terms of u, making it easier to evaluate. The differential dx is also transformed according to the substitution, allowing us to rewrite the integral in a more manageable form.
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Substitution With an Extra Variable

Differentiation as a Check

Differentiation is the process of finding the derivative of a function, which can be used to verify the correctness of an integral. After evaluating an indefinite integral, differentiating the result should yield the original integrand. This serves as a crucial check to ensure that the integration process was performed correctly.
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Determining Differentiability Graphically

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration (C) because the process of integration can yield multiple functions differing by a constant. Understanding the properties of indefinite integrals is essential for solving problems involving antiderivatives and applying techniques like substitution.
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Introduction to Indefinite Integrals
Related Practice
Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dt βˆ«β‚€α΅— d𝓍/(1 + 𝓍²) + βˆ«β‚ΒΉ/α΅— dx/(1 + 𝓍²)

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Textbook Question

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ’ on [a,b]. Identify Ζ’ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   βˆ‘ (𝓍ₖ*Β² + 1) βˆ†π“β‚– on [0,2]                                                                                                                                                                            

  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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Textbook Question

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ’ and the 𝓍-axis. Evaluate the following integrals.



βˆ«β‚€α΅ƒ Ζ’(𝓍) d𝓍

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Textbook Question

General results Evaluate the following integrals in which the function Ζ’ is unspecified. Note that ƒ⁽ᡖ⁾ is the pth derivative of Ζ’ and Ζ’α΅– is the pth power of Ζ’. Assume Ζ’ and its derivatives are continuous for all real numbers. 

∫ (5 Ζ’Β³ (𝓍) + 7Ζ’Β² (𝓍) + Ζ’ (𝓍 )) Ζ’'(𝓍) d𝓍

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Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 βˆ«β‚‹β‚ΒΉ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

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Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 βˆ« (sin⁡ 𝓍 + 3 sinΒ³ 𝓍― sin 𝓍) cos 𝓍 d𝓍

74
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