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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 5.5.79
Chapter 5, Problem 5.5.79

Variations on the substitution method Evaluate the following integrals.                                                                                                        
                                                                                                                                                                    
 βˆ« π“/(βˆšπ“β€•4) d𝓍

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Step 1: Identify the substitution method. Let u = βˆšπ“ - 4. This substitution simplifies the square root term in the denominator.
Step 2: Differentiate u with respect to 𝓍 to find du. Since u = βˆšπ“ - 4, differentiate both sides to get du = (1/(2βˆšπ“)) d𝓍.
Step 3: Rewrite the integral in terms of u. Substitute βˆšπ“ = u + 4 and d𝓍 = 2βˆšπ“ du into the integral. This transforms the integral into a simpler form.
Step 4: Simplify the integral. Replace βˆšπ“ in the numerator with u + 4, and simplify the expression to make it easier to integrate.
Step 5: Integrate the simplified expression with respect to u. After integration, substitute back u = βˆšπ“ - 4 to return to the original variable 𝓍.

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Key Concepts

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

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. This involves selecting a new variable, often denoted as 'u', which is a function of the original variable. By substituting 'u' into the integral, the integrand can often be transformed into a simpler form, making it easier to evaluate the integral.
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Definite vs. Indefinite Integrals

Integrals can be classified as definite or indefinite. An indefinite integral represents a family of functions and includes a constant of integration, while a definite integral computes the area under the curve between two specified limits. Understanding the difference is crucial for applying the correct evaluation techniques and interpreting the results accurately.
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Definition of the Definite Integral

Rational Functions and Their Integration

Rational functions are ratios of polynomials, and their integration often requires specific techniques, such as partial fraction decomposition or substitution. In the given integral, the presence of a square root in the denominator suggests that a substitution may simplify the expression, allowing for easier integration. Recognizing the form of the rational function is key to selecting the appropriate method.
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Related Practice
Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = xΒ³ on [3,11] with n = 32.

Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  βˆ«β‚€^Ο€/Β² (cos ΞΈ sin ΞΈ) / √(cosΒ² ΞΈ + 16) dΞΈ (Hint: Begin with u = cos ΞΈ .)

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Textbook Question

Suppose an object moves along a line at 15 m/s, for 0 ≀ t < 2 and at 25 m/s, for 2 ≀ t ≀ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≀ t ≀ 5.

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Textbook Question

Explain why βˆ«β‚α΅‡ Ζ’ β€²(𝓍) d𝓍 = Ζ’(b) ― Ζ’(a)

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


βˆ«β‚βΉ 2/(βˆšπ“) d𝓍

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Textbook Question

When using a change of variables u = g(𝓍) to evaluate the definite integral βˆ«β‚α΅‡ Ζ’(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?