Textbook Question
95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.6.33
7–84. Evaluate the following integrals.
33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0
Verified step by step guidance1
Step 1: Recognize that the integral involves a rational function with an exponential term in the numerator and a sum of squares in the denominator. This suggests a substitution method might simplify the problem.
Step 2: Let u = eˣ. Then, differentiate u with respect to x to find du: du = eˣ dx. This substitution replaces eˣ dx in the integral with du.
Step 3: Rewrite the integral in terms of u. Since u = eˣ, the denominator becomes a² + u², and the integral becomes ∫ [u / (a² + u²)] du.
Step 4: Recognize that the new integral ∫ [u / (a² + u²)] du can be simplified using the derivative of the denominator. Specifically, the derivative of a² + u² with respect to u is 2u, which suggests a natural logarithm substitution.
Step 5: Apply the substitution and simplify the integral. The result will involve a logarithmic function, specifically ln(a² + u²), and then substitute back u = eˣ to express the solution 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.
Integration Techniques
Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fractions. In this case, recognizing the form of the integrand, particularly the presence of the exponential function and the polynomial in the denominator, is crucial for selecting the appropriate technique to simplify the integral.
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Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions, such as eˣ, are functions of the form f(x) = a^x, where 'a' is a constant. They have unique properties, including the fact that their derivatives and integrals are also exponential functions. Understanding how to manipulate these functions is essential for evaluating integrals involving them, especially when combined with other algebraic expressions.
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6:13Exponential Functions
Improper Integrals
Improper integrals occur when the limits of integration are infinite or when the integrand approaches infinity within the interval of integration. In this problem, recognizing whether the integral converges or diverges is important, as it affects the evaluation process. Techniques such as limits can be applied to handle these cases effectively.
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11:11Improper Integrals: Infinite Intervals
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.
31. ∫ √(x² - 8x) dx, x > 8
Textbook Question
63. (Use of Tech) Normal distribution of heights
The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral
(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.
What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx
Textbook Question
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
33. ∫(0 to π) sin x cos(3x) dx = 0
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Textbook Question
87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.
A: dx = 2/(1 + u²) du
B: sin x = 2u/(1 + u²)
C: cos x = (1 - u²)/(1 + u²)
91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).