In this paper, through a direct computation with subintervals partitioning [0, 1], we compute better a posteriori bounds for the average case error of the difference between the true value of $I(f)=int_{0}^{1}f(x)dx$ with $f{in}C^r$[0, 1] minus the composite trapezoidal rule and the composite trapezoidal rule minus the basic trapezoidal rule for $r{geq}3$ by using zero mean-Gaussian.