Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 8, Problem 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.
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx
Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to 2 of (dx / (1 - x))
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
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Textbook Question
Evaluate the integrals in Exercises 53–58.
∫ sin(2x) cos(3x) dx