Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.7.38
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.
Guida verificata passo dopo passo1
Recognize that the integral given is \( \int_0^1 \frac{1}{1 + x^2} \, dx \), which is related to the arctangent function and is used to estimate \( \pi \) by the formula \( \pi = 4 \times \int_0^1 \frac{1}{1 + x^2} \, dx \).
Choose a numerical integration method to approximate the integral, such as the Trapezoidal Rule, Simpson's Rule, or a numerical integration function on a calculator or computer.
Divide the interval \([0,1]\) into \(n\) subintervals of equal width \( \Delta x = \frac{1 - 0}{n} = \frac{1}{n} \).
Calculate the function values \( f(x) = \frac{1}{1 + x^2} \) at the endpoints and at each subinterval point \( x_i = i \Delta x \) for \( i = 0, 1, 2, ..., n \).
Apply the chosen numerical integration formula to approximate the integral \( \int_0^1 \frac{1}{1 + x^2} \, dx \), then multiply the result by 4 to estimate \( \pi \).
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Definite Integral
A definite integral calculates the accumulated area under a curve between two limits. In this problem, the integral from 0 to 1 of 1/(1 + x²) represents the area under the curve of the function 1/(1 + x²), which is related to the arctangent function and helps approximate π.
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Percorso guidato
05:43
Definition of the Definite Integral
Numerical Integration
Numerical integration involves approximating the value of an integral using computational methods when an exact solution is difficult or unnecessary. Techniques like the trapezoidal rule or Simpson's rule estimate the area under a curve by summing areas of simpler shapes, useful for evaluating integrals like the one given.
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6:47Finding Limits Numerically and Graphically
Relationship Between Integral and π
The integral of 1/(1 + x²) from 0 to 1 equals arctan(1), which is π/4. Multiplying this integral by 4 yields π. This connection allows π to be approximated by evaluating the integral numerically, linking calculus with fundamental constants.
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05:23Finding Area Between Curves on a Given Interval
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