Evaluate the definite integral \(J=\int_{0}^{1} \frac{d x}{1+x^{2}}\) using:

(a) an exact analytic formula obtained from calculus;

(b) approximate methods: (i) the trapezoidal rule \((n=4)\); (ii) Simpson's rule \((2 m=4)\); and (iii) the Gaussian integration formula \((n=4)\).

(c) Compute the error bounds for the trapezoidal rule in part (b).

(d) Compare the results obtained in parts (a) and (b).