No output for N[Limit[Integrate[Sin[x]^n/(Sin[x]^n + Cos[x]^n), {x, 0, Pi/2}], n -> Infinity]]

Question

Is my input below correct? I've received no response from Mathematica.

N[Limit[Integrate[Sin[x]^n/(Sin[x]^n + Cos[x]^n), {x, 0, Pi/2}], n -> Infinity]]

Thank you in advance for your feedback!

Answer 1

Not an answer, just will not fit in the comment.

Mathematica can't seem to be able to integrate Sin[x]^n/(Sin[x]^n + Cos[x]^n analytically. I tried using the formulas for Sin[x]^n and Cos[x]^n for odd n with the hope it will help. But still no luck. Using Wiki, the formulas are from here

mysin[x_,n_] := (2/(2^n)) Sum[(-1)^((n - 1)/2 - k) Binomial[n, k]*
     Sin[(n - 2 k) x], {k, 0, (n - 1)/2}];

mycos[x_,n_] := (2/(2^n)) Sum[Binomial[n, k]*Cos[(n - 2 k) x], {k, 0, (n - 1)/2}];

integrand = 
 Assuming[Element[n, Integers] && n > 0 && Element[x, Reals], 
  FullSimplify[mysin[x, n]/(mysin[x, n] + mycos[x, n])]];

res = ComplexExpand[integrand];
Integrate[res, {x, 0, Pi/2}]

Its been running for 30 minutes.

But doing numerical integration

Table[
 NIntegrate[Sin[x]^n/(Sin[x]^n + Cos[x]^n), {x, 0, Pi/2}], {n, 0, 10}]

(*Out[1]= {0.785398, 0.785398, 0.785398, 0.785398, 0.785398, 0.785398, \
0.785398, 0.785398, 0.785398, 0.785398, 0.785398}*)

So may be you do not not need to do analytical integration after all? Are you sure this integrand can be integrated analytically?