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Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.2.7
Capitolo 8, Problema 8.2.7

Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.
7. ∫ (sec²x) · ln(tan x + 2) dx

Guida verificata passo dopo passo
1
Identify a substitution that simplifies the integral. Notice that the argument of the logarithm is \( \tan x + 2 \), and the derivative of \( \tan x \) is \( \sec^2 x \), which appears as a factor in the integral. So, let \( u = \tan x + 2 \).
Compute the differential \( du \) in terms of \( dx \). Since \( u = \tan x + 2 \), then \( \frac{du}{dx} = \sec^2 x \), which implies \( du = \sec^2 x \, dx \).
Rewrite the integral in terms of \( u \) and \( du \). The integral \( \int (\sec^2 x) \cdot \ln(\tan x + 2) \, dx \) becomes \( \int \ln u \, du \).
Recall the formula for the integral of \( \ln x \): \( \int \ln x \, dx = x \ln x - x + C \). Use this formula to evaluate \( \int \ln u \, du \).
After integrating, substitute back \( u = \tan x + 2 \) to express the answer in terms of \( x \).

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Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves identifying a part of the integrand as a new variable u, then rewriting the integral in terms of u and du. This technique is especially useful when the integral contains a composite function.
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Euler's Method

Integral of ln x

The integral of the natural logarithm function, ∫ ln x dx, can be evaluated using integration by parts. The formula is ∫ ln x dx = x ln x - x + C. This result is essential when the integral reduces to a form involving ln u, allowing direct evaluation after substitution.
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Integrals of Natural Exponential Functions (e^x)

Derivative of tan x and sec² x

Recognizing that the derivative of tan x is sec² x is crucial for substitution in this problem. Since sec² x dx equals d(tan x), it allows the integral involving sec² x and ln(tan x + 2) to be rewritten in terms of u = tan x + 2, facilitating the reduction to ∫ ln u du.
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Derivative of the Natural Exponential Function (e^x)
Pratica correlata
Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

37. ∫ dx / √(x² + 10x), x >

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Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

11. ∫ 3u / (2u + 7) du

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Domanda del libro di testo

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C


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f(x) = 1/(x√(x² - 36)), [12/√3 , 12]

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Domanda del libro di testo

92–98. Evaluate the following integrals.

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Domanda del libro di testo

49–52. {Use of Tech} Simpson’s Rule

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Domanda del libro di testo

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

44. ∫ 1/√(16 + 4x²) dx

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