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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 5.5.23
Chapter 5, Problem 5.5.23

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 βˆ« 𝓍³ (𝓍⁴ + 16)⁢ d𝓍

Verified step by step guidance
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Step 1: Identify the substitution. Since the integral involves a composite function (𝓍⁴ + 16) raised to a power, let u = 𝓍⁴ + 16. This substitution simplifies the integral by reducing the complexity of the expression.
Step 2: Compute the derivative of u with respect to 𝓍. Differentiating u = 𝓍⁴ + 16 gives du/d𝓍 = 4𝓍³. Rearrange to express du in terms of d𝓍: du = 4𝓍³ d𝓍.
Step 3: Rewrite the integral in terms of u. Substitute u = 𝓍⁴ + 16 and du = 4𝓍³ d𝓍 into the original integral. The integral becomes (1/4) ∫ u⁢ du, where the factor of 1/4 accounts for the 4𝓍³ in du.
Step 4: Integrate with respect to u. Use the power rule for integration: ∫ u⁢ du = (u⁷)/7. Thus, the integral becomes (1/4) * (u⁷)/7.
Step 5: Substitute back u = 𝓍⁴ + 16 into the result. Replace u in the expression with the original variable to return to the original terms: (1/28) * (𝓍⁴ + 16)⁷ + C, where C is the constant of integration.

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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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Introduction to Indefinite Integrals

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. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be accurate. This step is crucial in calculus as it ensures that the integration process was performed correctly and helps identify any potential errors in the calculations.
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Related Practice
Textbook Question

Use geometry and properties of integrals to evaluate


βˆ«β‚€ΒΉ (2𝓍 + √(1―𝓍²) + 1) d𝓍

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

Ζ’(𝓍) = 𝓍³ on [―1, 1]

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Textbook Question

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.

The region between the graph of y = 1 - |x| and the x-axis, for -2 ≀ x ≀ 2

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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‚β° Ζ’(𝓍) d𝓍

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.                       

                                                                                                                                                                                       

 βˆ«β‚€β΅ (𝓍²―9) d𝓍 

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« 𝓍eΛ£Β² d𝓍