Math Problem Statement

Given integral |ln(x)|^2 dx, use 8 sub intervals to find the Tn approximation. Round to 6 decimal places

Solution

To approximate the integral of (ln(x))2\left( \ln(x) \right)^2 over a given interval using the Trapezoidal Rule with 8 subintervals, we need to know the interval over which the integral is to be calculated. The Trapezoidal Rule formula is:

Tn=Δx2[f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn)]T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]

Where:

  • n=8n = 8 (the number of subintervals),
  • Δx=ban\Delta x = \frac{b - a}{n} is the width of each subinterval,
  • f(x)=(ln(x))2f(x) = \left( \ln(x) \right)^2.

Let me know the specific interval [a,b][a, b] for the integral so I can proceed with the calculation!

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Math Problem Analysis

Mathematical Concepts

Numerical Integration
Trapezoidal Rule
Integral of Natural Logarithm

Formulas

Trapezoidal Rule formula: T_n = (Δx/2) [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_(n-1)) + f(x_n)]
Δx = (b - a) / n
f(x) = (ln(x))^2

Theorems

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Suitable Grade Level

Undergraduate