How to fix the Y-axis and normalize density?

Question

I want to plot a density curve but when I draw the y-axis not getting to 1 (as normalized)
  Subscript[a, p]=0.428;Subscript[[Alpha], p] =1.04;R = 2.45;

Subscript[[Rho], p1][r_] = Subscript[[Rho], 0]/((E^((r - R)/Subscript[a, p]) + E^(-((r + R)/Subscript[a, p])))*(0.5*(r/R)^2 + 0.5)^Subscript[[Alpha], p] + 1)

(I used this code to find Subscript[[Rho], 0] and make the density normalized)
  4*Pi*NIntegrate[Subscript[[Rho], p1][r]*r^2, {r, 0, Infinity}]

Plot[Subscript[[Rho], p1][r], {r, 0, 8}]

JimB · Answer

You need to estimate the value of $rho_0$ using NIntegrate:
Subscript[a, p] = 0.428; Subscript[α, p] = 1.04; R = 2.45;

Subscript[ρ, 0] = 1/NIntegrate[1/(
   1 + (E^(2.336448598130841` (-2.45` + r)) + 
       E^(-2.336448598130841` (2.45` + r))) (0.5` + 
       0.08329862557267803` r^2)^1.04`), {r, 0, ∞}]
Subscript[ρ, p1][r_] = Subscript[ρ, 0]/((E^((r - R)/Subscript[a, p]) + 
       E^(-((r + R)/Subscript[a, p])))*(0.5*(r/R)^2 + 0.5)^Subscript[α, p] + 1)

Plot[Subscript[ρ, p1][r], {r, 0, 8}]

And in Mathematica you should avoid superscripts and subscripts.  Using ap and αp would be less prone to unexpected consequences.

How to fix the Y-axis and normalize density?

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