Problem on the normal-cauchy bayes estimator

Question

Solve the following problem:

For the normal-Cauchy Bayes estimator

δ(x) = ∫^∞ _-∞  (θ/1+θ²)e^-(x-θ)2/2 dθ  /  ∫^∞ _-∞  (1/1+θ²)e^-(x-θ)2/2dθ

solve the following questions when x = 0, 2, 4.

a. Plot the integrands, and use Monte Carlo integration based on a Cauchy simulation to calculate the integrals.

b. Monitor the convergence with the standard error of the estimate. Obtain three digits of accuracy with probability .95.

c. Repeat the experiment with a Monte Carlo integration based on a normal simulation and compare both approaches.