IGNOU MTE 10 2018 - Compute the values of by using the trapezoidal rule with h = 0.5,

Question

Compute the values of $1=\int_{a}^{1}\frac{dx}{1+x^2}$

by using the trapezoidal rule with h = 0.5, 0.25, 0.125 . Improve this value by using the Romberg’s method. Compare your result with the true value.

14 Feb 2018

Answer :

Word Count : 351

To solve this problem numerically, we need to compute the integral: \[ \int_{a}^{1} \frac{dx}{1 + x^2} \] using the trapezoidal rule with different values of $h = 0.5, 0.25, 0.125$. Then, we'll improve the result using Romberg's method and compare it with the true value. ### Step 1: Understanding the Integral The integral $ \int \frac{dx}{1 + x^2} $ is a standard integral and its solution is: \[ \int \frac{dx}{1 + x^2} = \tan^{-1}(x) \] Thus, the

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