Domanda del libro di testo
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)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 8.5.26
23-64. Integration Evaluate the following integrals.
26. ∫₀¹ [1 / (t² - 9)] dt
Guida verificata passo dopo passo1
Identify the integral to be evaluated: \(\int_0^1 \frac{1}{t^2 - 9} \, dt\).
Recognize that the denominator can be factored using the difference of squares: \(t^2 - 9 = (t - 3)(t + 3)\).
Use partial fraction decomposition to express \(\frac{1}{t^2 - 9}\) as \(\frac{A}{t - 3} + \frac{B}{t + 3}\), and solve for constants \(A\) and \(B\).
Rewrite the integral as the sum of two simpler integrals: \(\int_0^1 \frac{A}{t - 3} \, dt + \int_0^1 \frac{B}{t + 3} \, dt\).
Integrate each term separately using the natural logarithm function: \(\int \frac{1}{t - a} \, dt = \ln|t - a| + C\), then apply the limits from 0 to 1.
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Improper Integrals
An improper integral occurs when the integrand is undefined or unbounded within the interval of integration. In this problem, the denominator t² - 9 equals zero at t = ±3, which lies outside the interval [0,1], so the integral is proper. However, understanding improper integrals helps in evaluating integrals with singularities.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Partial Fraction Decomposition
Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. For the integrand 1/(t² - 9), factor the denominator as (t - 3)(t + 3) and express the fraction as a sum of terms with linear denominators, facilitating straightforward integration.
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Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Definite Integration Techniques
Definite integration involves finding the exact area under a curve between two limits. After decomposing the integrand, integrate each term separately and then apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits to find the integral's value.
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Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
7–84. Evaluate the following integrals.
11. ∫ from 0 to π/4 (sec x – cos x)² dx
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Domanda del libro di testo
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.
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Domanda del libro di testo
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.
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Domanda del libro di testo
59. Perpetual Annuity
Imagine that today you deposit $B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of $I per year from the account forever. The amount of money that must be deposited is:
B = I × ∫(from 0 to ∞) e^(-rt) dt
where r = p/100.
Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of \$5000 per year. How much must you deposit today?
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Domanda del libro di testo
9–61. Trigonometric integrals Evaluate the following integrals.
22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx
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