Domanda del libro di testo
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.
∫₀⁵ (𝓍²―9) d𝓍
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Ch. 5 - Integration
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 5, Problema 5.5.66
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀ᵉ² (ln p)/p dp
Guida verificata passo dopo passo1
Step 1: Recognize that the integral ∫₀ᵉ² (ln p)/p dp can be solved using a substitution method. Let u = ln(p), which simplifies the logarithmic term.
Step 2: Compute the derivative of u with respect to p. Since u = ln(p), we have du/dp = 1/p, or equivalently, du = (1/p) dp.
Step 3: Substitute u and du into the integral. The integral becomes ∫₀ᵉ² (ln p)/p dp = ∫₀ᵉ² u du.
Step 4: Adjust the limits of integration to match the substitution. When p = 0, ln(p) is undefined, but the integral is defined for positive values approaching 0. When p = e², ln(p) = 2. Thus, the new limits are from u = 0 to u = 2.
Step 5: Evaluate the integral ∫₀² u du using the formula for the integral of u, which is (u²/2). Apply the limits of integration to find the result.
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Definite Integrals
A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, over the interval [a, b].
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Percorso guidato
05:43
Definition of the Definite Integral
Change of Variables
Change of variables, or substitution, is a technique used in integration to simplify the integrand by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The Jacobian determinant is often used to adjust for the change in variable when dealing with multiple integrals.
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Percorso guidato
06:35Changing Geometries
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is a fundamental function in calculus, particularly in integration and differentiation. The natural logarithm has unique properties, such as ln(ab) = ln(a) + ln(b), which can be useful when simplifying integrals involving logarithmic expressions.
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05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
Areas of regions Find the area of the following regions.
The region bounded by the graph of ƒ(𝓍) = x /√(𝓍² ―9) and the 𝓍-axis between and 𝓍 = 4 and 𝓍= 5
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Domanda del libro di testo
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍
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