Domanda del libro di testo
23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.1.69
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
Guida verificata passo dopo passo1
Step 1: Identify the substitution given in the problem. Here, we are instructed to use u = sec(x). This substitution will help simplify the integral.
Step 2: Compute the derivative of u with respect to x. Since u = sec(x), we have du/dx = sec(x)tan(x). Therefore, du = sec(x)tan(x) dx.
Step 3: Rewrite the integral ∫(tan(x) sec²(x)) dx using the substitution u = sec(x). Replace sec(x) with u and tan(x) sec(x) dx with du.
Step 4: After substitution, the integral becomes ∫u du because sec²(x) becomes u² and tan(x) sec(x) dx is replaced by du.
Step 5: Integrate ∫u du to find the antiderivative. The result will be expressed in terms of u, and then substitute back u = sec(x) to express the final answer in terms of x.
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Integration
Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for solving problems involving accumulation and area, and it often requires techniques such as substitution to simplify the integrand.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Substitution Method
The substitution method is a technique used in integration to simplify complex integrals. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. In this case, using the substitution u = sec(x) allows us to express the integral in terms of u, making it easier to evaluate.
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07:33Euler's Method
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving integrals involving trigonometric functions. In this problem, recognizing the relationship between tan(x), sec(x), and their derivatives is key to applying the substitution effectively.
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7:17Verifying Trig Equations as Identities
Pratica correlata
Domanda del libro di testo
37-40. {Use of Tech} Temperature data
Howdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.
Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:
T_avg = (1/12) × ∫(0 to 12) T(t) dt
38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.
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Domanda del libro di testo
65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
67. ∫tan⁴(3y) dy
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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.
28. ∫ ln² x dx
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Domanda del libro di testo
54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:
55. ∫ x² cos(5x) dx
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Domanda del libro di testo
9–40. Integration by parts Evaluate the following integrals using integration by parts.
32. ∫ from 0 to 1 x² 2ˣ dx
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