Compute the variance of the logistic distribution π2 /3 = 2 ∞ 0 x2e−x(1+ e−x )−2 dx to five decimal digits using a simple quadrature method. The first step is to bound the tail by 2  ∞ t x2e−x dx by...

Compute the variance of the logistic distribution π2 /3 = 2 ∞ 0 x2e−x(1+ e−x )−2 dx to five decimal digits using a simple quadrature method. The first step is to bound the tail by 2  ∞ t x2e−x dx by finding an appropriate value of t (and proving the bound). Next find the appropriate value of h so that the integration error from (10.3.6) or Exercise 10.1 is small enough. Finally, integrate over [0, t] using the midpoint rule.