n = 60 ; df = n - 1 ; d = .5 ; t = d*sqrt(n) ## some observed data

 curve( dt(x*sqrt(n), df, t)*sqrt(n), -.2, 1.2) ## The sampling dist. of "d"


f <- function (ncp, alpha, q, df) {  ## Method 1 as stated in the stats TEXTS
  abs(suppressWarnings(pt(q = t, df = df, ncp, lower.tail = FALSE)) - alpha)
 }

CI1 = sapply(c(0.025, 0.975),
function(x) optimize(f, interval = c(-37, 37), alpha = x, q = t, df = df, tol = 1e-10)[[1]]/sqrt(n))


CI2 = qt( c(.025, .975), df, ncp = t) / sqrt(n)  ## Method 2

text(CI1, 0, round(CI1, 4), font = 2)
text(CI2, 0, round(CI2, 4), font = 2, pos = 3, col = 2)

integrate(function(x) dt(x*sqrt(n), df, t)*sqrt(n), -Inf, CI1[1])[[1]]
integrate(function(x) dt(x*sqrt(n), df, t)*sqrt(n), -Inf, CI2[1])[[1]]