How to plot the result of NIntegrate when integrand has parameters?

Question

I am trying to plot f as a function of k (n,k naturals):

f[n_, k_] := NIntegrate[( n)^(2 k)g1[n, k, \[Theta]], {\[Theta], -\[Pi], \[Pi]}]/NIntegrate[g2[n, k, \[Theta]], {\[Theta], -\[Pi], \[Pi]}];

g1[n_, k_, \[Theta]_] := (Sin[(n \[Theta])/2]/Sin[\[Theta]/2])^(
   2 k) (2 Sin[\[Theta]/2])^(2 k);

g2[n_, k_, \[Theta]_] := (Sin[(n \[Theta])/2]/Sin[\[Theta]/2])^(2 k);

n=10;
Plot[f[n, k], {k, 1, 5}]
```

Answer 1

We can use NumericQ and ReIm to plot this function. Also, it is better to divide f into two parts:

f[n_?NumericQ, k_?NumericQ] := 
  NIntegrate[(n)^(2 k) g1[n, 
     k, \[Theta]], {\[Theta], -\[Pi], \[Pi]}];
f1[n_?NumericQ, k_?NumericQ] := 
  NIntegrate[g2[n, k, \[Theta]], {\[Theta], -\[Pi], \[Pi]}];
g1[n_, k_, \[Theta]_] := (Sin[(n \[Theta])/2]/
      Sin[\[Theta]/2])^(2 k) (2 Sin[\[Theta]/2])^(2 k);

g2[n_, k_, \[Theta]_] := (Sin[(n \[Theta])/2]/Sin[\[Theta]/2])^(2 k);

n = 10;

Plot[Evaluate[ReIm[f[n, k]/f1[n, k]]], {k, 1, 2}, Frame -> True]

Answer 2

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

g1[n_, k_, θ_] := (Sin[(n θ)/2]/
      Sin[θ/2])^(2 k) (2 Sin[θ/2])^(2 k);

g2[n_, k_, θ_] := (Sin[(n θ)/2]/Sin[θ/2])^(2 k);

The natural numbers is an ambiguous constraint, it can mean either positive or nonnegative integers. I have defined f for n positive and k nonnegative.

f[n_Integer?Positive, k_Integer?NonNegative] :=
  NIntegrate[(n)^(2 k) g1[n, k, θ], {θ, -π, π}]/
   NIntegrate[g2[n, k, θ], {θ, -π, π}];

Manipulate[
 DiscretePlot[f[n, k], {k, 0, 4},
  AxesLabel -> {k, f},
  PlotRange -> All],
 {{n, 5}, Range[5]}]