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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 5.5.20
Chapter 5, Problem 5.5.20

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 βˆ« [(βˆšπ“ + 1)⁴ / 2βˆšπ“ d𝓍

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Step 1: Identify the substitution that simplifies the integral. Let u = βˆšπ“ + 1. Then, differentiate u with respect to 𝓍 to find du/d𝓍. Since u = βˆšπ“ + 1, we have du/d𝓍 = 1/(2βˆšπ“). Rearrange to express d𝓍 in terms of du: d𝓍 = 2βˆšπ“ du.
Step 2: Rewrite the integral in terms of u. Substitute βˆšπ“ + 1 with u and d𝓍 with 2βˆšπ“ du. The integral becomes ∫ [(u⁴) / (2βˆšπ“)] * (2βˆšπ“ du). Notice that the βˆšπ“ terms cancel out, simplifying the integral to ∫ u⁴ du.
Step 3: Integrate u⁴ with respect to u. Use the power rule for integration: ∫ u⁴ du = (u⁡)/5 + C, where C is the constant of integration.
Step 4: Substitute back the original variable 𝓍 into the solution. Recall that u = βˆšπ“ + 1, so replace u in the result with βˆšπ“ + 1. The integral becomes ((βˆšπ“ + 1)⁡)/5 + C.
Step 5: Verify the solution by differentiating the result. Differentiate ((βˆšπ“ + 1)⁡)/5 + C with respect to 𝓍 and confirm that it simplifies back to the original integrand [(βˆšπ“ + 1)⁴ / 2βˆšπ“]. This ensures the solution is correct.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.
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Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex expressions, allowing for easier integration and ultimately leading to the correct antiderivative.
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Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the antiderivative found corresponds to the original integrand. If the derivative of the antiderivative matches the integrand, it confirms that the integration was performed correctly, reinforcing the relationship between differentiation and integration.
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Related Practice
Textbook Question

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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.                                                                                  

                                                                                                                                                                    

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Textbook Question

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