Hello Alex. Not life in general, just mine. It's from the Last Samurai with Tom Cruise. Hey, it's not hard to check that as we can numerically integrate the inverse directly: Just do a table of {f(x),x}, generate an "interpolation" function on that table, bingo-bango (the code is Mathematica):

Code:

In[83]:=
f[x_] := (1/2)*x + (3/4)*x^(1/2);
mytable = Table[{f[x], x},
{x, 0, 10, 0.1}];
myinverse = Interpolation[mytable];
rval = 4;
kval = 2;
i1 = NIntegrate[t^2*myinverse[kval*t],
{t, 0, rval}]
p1 = First[x /. Solve[8 == (1/2)*x +
(3/4)*x^(1/2), x]];
i2 = NIntegrate[(1/kval^3)*
((1/2)*u + (3/4)*u^(1/2))^2*u*
(1/2 + 3/8/u^2^(-1)), {u, 0, p1}]
Out[88]=
167.62359195110528
Out[90]=
167.62360953565704