Getting Integrate::ilim error while plotting

Question

I am a beginner in Mathematica. I want to plot for

t = 5

f[x_] = Integrate[Sqrt[ Tanh[x]], x]

r[x_] = Integrate[Cos[f[x]], x]

Plot[r[x], {x, -t, t}, 
  AxesLabel -> {"x"}, 
  PlotLabel -> "y", 
  PlotRange -> All, 
  PlotPoints -> 500]

but I'm getting error messages:

Integrate::ilim: Invalid integration variable or limit(s) in -4.99998. >>
Integrate::ilim: Invalid integration variable or limit(s) in -4.97994. >>
Integrate::ilim: Invalid integration variable or limit(s) in -4.9599. >>

Anyone knows a solution?

Answer 1

If you had looked at the output of

r[x_] = Integrate[Cos[f[x]], x]

you would have seen that Mathematica returned the integral unevaluated because it couldn't find a closed-form solution. One way to proceed with solving your problem is to numerically integrate to make a table of points along r[x] in the interval of interest.

f[x_] = Integrate[Sqrt[Tanh[x]], x]
With[{t = 5., dt = .05},
  data = Table[{u, NIntegrate[Cos[f[x]], {x, -t, u}]}, {u, -t, t, dt}];
  ListLinePlot[data, AxesLabel -> {"x", "y"}]]