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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 5.5.18
Chapter 5, Problem 5.5.18

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

Verified step by step guidance
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Step 1: Recognize that the integral βˆ«π“eΛ£Β² d𝓍 suggests a substitution method because the derivative of the inner function xΒ² appears as a factor (𝓍). This is a common pattern for substitution.
Step 2: Let u = xΒ². Then, compute the derivative of u with respect to x: du/dx = 2x, or equivalently, du = 2x dx.
Step 3: Rewrite the integral in terms of u. Substitute u = x² and du = 2x dx into the integral. This transforms the integral into (1/2)∫eᡘ du, where the factor of 1/2 comes from adjusting for the 2x in du.
Step 4: Evaluate the integral ∫eᡘ du. The antiderivative of eᡘ is simply eᡘ, so the integral becomes (1/2)eᡘ + C, where C is the constant of integration.
Step 5: Substitute back u = xΒ² to express the result in terms of x. The final expression is (1/2)eΛ£Β² + C. To verify, differentiate this result with respect to x and confirm that 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.

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, where the goal is to determine a function F(x) such that F'(x) equals the integrand.
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Introduction to Indefinite Integrals

Change of Variables

The change of variables technique, also known as substitution, is a method 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 technique is particularly useful when dealing with composite functions or when the integrand contains products of functions.
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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 correct. This step is crucial in calculus as it ensures that the integration process was performed accurately.
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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

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

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of Ζ’(𝓍) = x /√(𝓍² ―9) and the 𝓍-axis between and 𝓍 = 4 and π“= 5

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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‚αΆœ Ζ’(𝓍) 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« 𝓍³ (𝓍⁴ + 16)⁢ d𝓍