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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.7
Chapter 8, Problem 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

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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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Key Concepts

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

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)
Related Practice
Textbook Question

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.

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Textbook Question

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.

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Textbook Question

{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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Textbook Question

4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.

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Textbook Question

92–98. Evaluate the following integrals.

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Textbook Question

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

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