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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.36
Capitolo 5, Problema 5.5.36

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ sec 4w tan 4w dw

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Step 1: Recognize the integral ∫ sec(4w) tan(4w) dw. This matches a standard form in calculus where the derivative of sec(x) is sec(x) tan(x).
Step 2: Use substitution to simplify the integral. Let u = 4w, which implies that du = 4 dw. Rewrite dw as dw = du/4.
Step 3: Substitute u into the integral. The integral becomes (1/4) ∫ sec(u) tan(u) du.
Step 4: Recall the standard result that the integral of sec(x) tan(x) is sec(x). Using this, the integral simplifies to (1/4) sec(u).
Step 5: Substitute back u = 4w to return to the original variable. The final expression is (1/4) sec(4w) + C, where C is the constant of integration.

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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

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 that can be simplified.
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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 has been performed accurately and helps identify any potential errors in the calculation.
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Pratica correlata
Domanda del libro di testo

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ [(x³ ― 4x) / (x² + 1)] dx 

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

                                                                                                                                                                              

 ∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍

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Domanda del libro di testo

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



∫₀ᵃ ƒ(𝓍) d𝓍

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Domanda del libro di testo

Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.

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General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers. 

∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍

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Identifying Riemann sums Fill in the blanks with an interval and a value of n.


4

∑ ƒ (1.5 + k) • 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]

k = 1

with n = ________ .

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