f(x) = + In(x) exp(x) + exp(-x) herex e [1, 5]. Use 5 nodes to uniformly cut the interval [1,5] into 4 sub- tervals. 1. Use Trapezoid rule to approximate f f(x)dx. 2. Use Simpson's rule to approximate i f(x)dx.
f(x) = + In(x) exp(x) + exp(-x) herex e [1, 5]. Use 5 nodes to uniformly cut the interval [1,5] into 4 sub- tervals. 1. Use Trapezoid rule to approximate f f(x)dx. 2. Use Simpson's rule to approximate i f(x)dx.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The given mathematical problem is about the function:
\[
f(x) = \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)} + \ln(x)
\]
where \( x \in [1, 5] \).
The task is to divide this interval \([1, 5]\) into 4 sub-intervals using 5 nodes. The instructions are as follows:
1. **Use the Trapezoid Rule** to approximate the integral \(\int_{1}^{5} f(x) \, dx\).
2. **Use Simpson’s Rule** to approximate the same integral \(\int_{1}^{5} f(x) \, dx\).
3. **Use 2 Gaussian points (\(n = 1\))** to evaluate the integration in each sub-interval, and then sum them to obtain the integral over the entire domain.
These steps involve numerical integration techniques that are commonly used to approximate definite integrals. The Trapezoid Rule and Simpson’s Rule are both based on particular polynomial approximations of the function \(f(x)\). Gaussian quadrature, on the other hand, is an advanced technique that uses strategically chosen sampling points and weights to improve the approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74ac81bd-2b81-497e-b1b6-4cc5e46be1b6%2F99e82b2b-b835-4b4a-9d02-09427eefb256%2Fjfr1nd5_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical problem is about the function:
\[
f(x) = \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)} + \ln(x)
\]
where \( x \in [1, 5] \).
The task is to divide this interval \([1, 5]\) into 4 sub-intervals using 5 nodes. The instructions are as follows:
1. **Use the Trapezoid Rule** to approximate the integral \(\int_{1}^{5} f(x) \, dx\).
2. **Use Simpson’s Rule** to approximate the same integral \(\int_{1}^{5} f(x) \, dx\).
3. **Use 2 Gaussian points (\(n = 1\))** to evaluate the integration in each sub-interval, and then sum them to obtain the integral over the entire domain.
These steps involve numerical integration techniques that are commonly used to approximate definite integrals. The Trapezoid Rule and Simpson’s Rule are both based on particular polynomial approximations of the function \(f(x)\). Gaussian quadrature, on the other hand, is an advanced technique that uses strategically chosen sampling points and weights to improve the approximation.
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