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In[177]:=
Clear[f,x]​​f[x_Real]=Simplify[x*(x**2+9./4.)/((x**2+3./2.+x*Sqrt[x**2+2.])*Sqrt[x**2+2.]),x>=0]​​Evaluate[f[5.]]​​Plot[f[x],{x,0.,5.},{PlotRange->Full,ImageSize->Medium}]​​Simplify[Nintegrate[f[x],{x,0.,5.}]]​​
Out[178]=
x(x**2+2.25)
x**2+2.
(
x**2+2.
x+x**2+1.5)
Out[179]=
5.(5.**2+2.25)
5.**2+2.
(5.**2+5.
5.**2+2.
+1.5)
Out[180]=
1
2
3
4
5
-1.0
-0.5
0.5
1.0
Out[181]=
Nintegrate(f(x),{x,0.,5.})