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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 5.5.12
Capitolo 5, Problema 5.5.12

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍

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1
Recognize that the integral involves an exponential function with a linear term in the exponent. To simplify, use the substitution u = 3𝓍 + 1, where a = 3 and b = 1.
Differentiate the substitution u = 3𝓍 + 1 with respect to 𝓍 to find du/d𝓍 = 3, or equivalently, du = 3 d𝓍.
Solve for d𝓍 in terms of du: d𝓍 = du/3. Substitute this and u into the integral, replacing 3𝓍 + 1 with u and d𝓍 with du/3.
The integral now becomes ∫e^u (du/3). Factor out the constant 1/3 to simplify the expression to (1/3) ∫e^u du.
Evaluate the integral of e^u, which is simply e^u. Replace u with the original substitution 3𝓍 + 1 to return to the variable 𝓍. The final result will include a constant of integration, C.

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Substitution Method in Integration

The substitution method is a technique used in calculus to simplify the process of integration. By substituting a part of the integrand with a new variable, it transforms the integral into a more manageable form. This method is particularly useful when dealing with composite functions, allowing for easier integration by changing the variable 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 of integration and include a constant of integration, typically denoted as 'C'. Understanding indefinite integrals is crucial for finding antiderivatives and solving differential equations, as they provide the general solution to integration problems.
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Introduction to Indefinite Integrals

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^(kx), where 'e' is the base of natural logarithms and 'k' is a constant. These functions are characterized by their rapid growth and unique properties, such as the fact that their derivative is proportional to the function itself. Recognizing the behavior of exponential functions is essential for evaluating integrals involving them, particularly in the context of substitution.
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{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 right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.

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Average distance on a triangle Consider the right triangle with vertices (0,0) ,(0,b) , and (a,0) , where a > 0 and b > 0. Show that the average vertical distance from points on the 𝓍-axis to the hypotenuse is b/2 , for all a > 0 .

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


∫¹₁/₂ (t⁻³ ― 8) dt

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Derivatives of integrals Simplify the following expressions.


d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍

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

                                                                                                                                                                              

 ∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍

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Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6

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