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 8.9.53
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
53. ∫ (from 0 to 1) ln x dx
Guida verificata passo dopo passo1
Recognize that the integral \( \int_0^1 \ln x \, dx \) is an improper integral because \( \ln x \) approaches \(-\infty\) as \( x \) approaches 0 from the right.
Rewrite the integral as a limit to handle the improper behavior at 0: \[ \int_0^1 \ln x \, dx = \lim_{t \to 0^+} \int_t^1 \ln x \, dx \].
Use integration by parts to evaluate \( \int \ln x \, dx \). Let \( u = \ln x \) and \( dv = dx \), then \( du = \frac{1}{x} dx \) and \( v = x \).
Apply the integration by parts formula: \[ \int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 \, dx = x \ln x - x + C \].
Evaluate the definite integral using the antiderivative: \[ \int_t^1 \ln x \, dx = [x \ln x - x]_t^1 = (1 \cdot \ln 1 - 1) - (t \ln t - t) \]. Then take the limit as \( t \to 0^+ \) to determine if the integral converges or diverges.
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Improper Integrals
Improper integrals involve integrals with infinite limits or integrands that approach infinity within the interval. To evaluate them, we use limits to handle points where the function is undefined or unbounded, determining if the integral converges or diverges.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Natural Logarithm Function Behavior near Zero
The natural logarithm function, ln(x), approaches negative infinity as x approaches zero from the right. This behavior creates an improper integral at the lower limit, requiring careful limit evaluation to determine convergence.
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05:18Derivative of the Natural Logarithmic Function
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation, useful for integrating products of functions. It transforms the integral into simpler parts, often applied when integrating functions like ln(x) that are difficult to integrate directly.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
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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
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv
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Domanda del libro di testo
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)
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Domanda del libro di testo
7–64. Integration review Evaluate the following integrals.
62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx
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