ClearAll

k=Sqrt[2]/2

f[x_]=1/(Sqrt[1-x^2]Sqrt[1-k^2x^2])

Plot[2*ArcSin[k*JacobiSN[t,k]],{t,0,20}]

Plot[Integrate[1/(Sqrt[1-x^2]Sqrt[1-k^2x^2]),{x,0,0.999}],{k,1/Sqrt[2],1}]

Integrate[1/(Sqrt[1-x^2]Sqrt[1-0.5*x^2]),{x,0,1}]

Plot[NIntegrate[1/(Sqrt[1-x^2]Sqrt[1-k^2x^2]),{x,0,0.999}],{k,1/Sqrt[2],1}]

FunctionDomain[Intgrate[(1/((Sqrt[1-(x^2)]) (Sqrt[1-0.5` (x^2)]))),{x,0,1}],x]

Number[Intgrate[(1/((Sqrt[1-(x^2)]) (Sqrt[1-0.5` (x^2)]))),{x,0,1}]]