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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 8.6.45
Chapter 8, Problem 8.6.45

7–84. Evaluate the following integrals.
45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

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Step 1: Recognize that the integral involves the function \( \frac{1}{(1 + e^x)^2} \). This suggests that substitution might simplify the problem. Look for a substitution that simplifies the denominator \( 1 + e^x \).
Step 2: Let \( u = 1 + e^x \). Then, differentiate \( u \) with respect to \( x \): \( \frac{du}{dx} = e^x \), or equivalently \( du = e^x dx \). Rewrite \( e^x dx \) in terms of \( u \).
Step 3: Substitute \( u \) into the integral. When \( x = 0 \), \( u = 1 + e^0 = 2 \). When \( x = \ln 2 \), \( u = 1 + e^{\ln 2} = 1 + 2 = 3 \). The integral becomes \( \int_{2}^{3} \frac{1}{u^2} \cdot \frac{du}{e^x} \). Since \( e^x = u - 1 \), replace \( e^x \) with \( u - 1 \).
Step 4: Simplify the integral to \( \int_{2}^{3} \frac{1}{u^2(u - 1)} du \). This integral can be solved using partial fraction decomposition. Express \( \frac{1}{u^2(u - 1)} \) as \( \frac{A}{u} + \frac{B}{u^2} + \frac{C}{u - 1} \), and solve for \( A \), \( B \), and \( C \).
Step 5: After finding the coefficients \( A \), \( B \), and \( C \), rewrite the integral as a sum of simpler integrals: \( \int \frac{A}{u} du + \int \frac{B}{u^2} du + \int \frac{C}{u - 1} du \). Evaluate each term separately and apply the limits of integration \( u = 2 \) to \( u = 3 \).

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Key Concepts

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

Definite Integrals

A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral from 0 to ln 2 indicates that we are interested in the area under the curve of the function 1 / (1 + eˣ)² from x = 0 to x = ln 2.
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Definition of the Definite Integral

Integration Techniques

To evaluate integrals, various techniques can be employed, such as substitution, integration by parts, or recognizing standard forms. For the given integral, a substitution involving the exponential function may simplify the process, allowing for easier integration of the function.
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Exponential Functions

Exponential functions, such as eˣ, are fundamental in calculus due to their unique properties, including their derivatives and integrals being proportional to the function itself. Understanding how to manipulate and integrate these functions is crucial for solving integrals involving eˣ, as seen in the integrand of the given problem.
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Related Practice
Textbook Question

79. Tabular integration extended Refer to Exercise 77.

a. The following table shows the method of tabular integration applied to

∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.

b. Solve the equation in part (a) for ∫ eʳ cos z dz.

c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).

Textbook Question

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

57. ∫ from 0 to π of (1 - cos2x)³ᐟ² dx

Textbook Question

23-64. Integration Evaluate the following integrals.

60.∫ 1/[(y² + 1)(y² + 2)] dy

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Textbook Question

7–64. Integration review Evaluate the following integrals.

60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

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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.

34. ∫ dx / (x(x¹⁰ + 1))