Increase Precision in Numerical Integration

Question

I need to generate a table of Chebyshev expansion coefficients of trigonometric functions (in this case Cos[2 Pi t] to very high accuracy. Code is:

ClearAll["Global`*"]

m = 4;

f[t] = Cos[2 Pi t];
Tn[t] = ChebyshevT[j, 2*t - 1];
wt[t] = 1/Sqrt[t - t^2];    

p = Table[
   NIntegrate[f[t]*Tn[t]*wt[t], {t, 0, 1}]/(Pi/2.0), {j, 0, m - 1}];
p[[1]] = p[[1]]/2;

p

Where f[t] is the trigonometric function, Tn[t], is the shifted Chebyshev polynomial, w[t] is the weight function. How can I increase the precision of this numerical integration?

I apologize if this is a duplicate, but I've read dozens of past questions dealing with $MinPrecision, $MinAccuracy, PrecisionGoal, WorkingPrecision, etc. and I've been unable to increase the precision. I'm stuck at machine precision still!

Answer 1

Your results are converted to MachinePrecision because you divide by the machine precision number Pi/2.0. You can get exact solutions with Integrate and use N. This will be more accurate than NIntegrate.

Block[{m = 4},
 p = Table[
    Integrate[f[t]*Tn[t]*wt[t], {t, 0, 1}]/(Pi/2), {j, 0, m - 1}];
 ]
p[[1]] = p[[1]]/2;

p
(*
  {-BesselJ[0, π],
   0,
   2 BesselJ[2, π],
   0}
*)

N[p, 20]
(*
  {0.30424217764409386420,
   0,
   0.97086786526301821941,
   0}
*)