Method III
Sin[Nx] ＝ Exp[iNx]－Exp[-iNx]／2i
Sin[x] ＝ Exp[ix]－Exp[-ix]／2i
Integrand
＝(Sin[Nx]／Sin[x])^2
＝(Exp[iNx]－Exp[-iNx]／Exp[ix]－Exp[-ix])^2
＝Σ(Exp[ix])^2r * (Exp[-ix])^2(N-r-1) r from 0~N-1
＝ΣExp[i*2(2r-N+1)x]
when 2r-N+1≠0
∫Exp[i*2(2r-N+1)x]dx＝0
when 2r-N+1＝0
∫Exp[i*2(2r-N+1)x]dx＝∫dx＝π
Hence,
Σ∫Exp[i*2(2r-N+1)x]dx＝Nπ
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