Domanda del libro di testo
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
9. ∫[5 to 5√3] √(100 - x²) dx
101
views
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 8.5.57
23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx
Guida verificata passo dopo passo1
Start by examining the integral \( \int \frac{x^{3} + 5x}{(x^{2} + 3)^{2}} \, dx \). Notice that the denominator is \( (x^{2} + 3)^{2} \), which suggests a substitution involving \( x^{2} + 3 \) might simplify the integral.
Let \( u = x^{2} + 3 \). Then, compute the differential \( du = 2x \, dx \), which implies \( x \, dx = \frac{du}{2} \). This substitution will help rewrite parts of the integral in terms of \( u \) and \( du \).
Rewrite the numerator \( x^{3} + 5x \) as \( x(x^{2} + 5) \). Using the substitution, express \( x^{3} + 5x = x(x^{2} + 5) = x(u - 3 + 5) = x(u + 2) \). This allows you to write the integral as \( \int \frac{x(u + 2)}{u^{2}} \, dx \).
Replace \( x \, dx \) with \( \frac{du}{2} \) from the substitution step, so the integral becomes \( \int \frac{u + 2}{u^{2}} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{u + 2}{u^{2}} \, du \).
Split the integral into simpler terms: \( \frac{1}{2} \int \left( \frac{u}{u^{2}} + \frac{2}{u^{2}} \right) du = \frac{1}{2} \int \left( \frac{1}{u} + 2u^{-2} \right) du \). Now, integrate each term separately using the power rule and logarithmic integration.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Integration of Rational Functions
This involves integrating functions expressed as the ratio of two polynomials. Techniques often include polynomial division, substitution, or partial fraction decomposition to simplify the integral into manageable parts.
Video consigliato:
6:04
Intro to Rational Functions
Substitution Method
A technique where a part of the integral is replaced with a new variable to simplify the expression. For example, setting u = denominator or a function inside the integral can transform the integral into a standard form.
Video consigliato:
07:33Euler's Method
Polynomial Division and Simplification
When the degree of the numerator is equal to or higher than the denominator, dividing polynomials can simplify the integrand. This step often precedes substitution or partial fractions to make integration straightforward.
Video consigliato:
07:00Taylor Polynomials
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.
46
views
Domanda del libro di testo
69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.
131
views
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
45
views
Domanda del libro di testo
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3
123
views
Domanda del libro di testo
7–84. Evaluate the following integrals.
35. ∫ from 0 to π/4 [(tan²θ + tanθ + 1) sec²θ] dθ
71
views