How can I combine NIntegrate with NDSolve?

Question

I have no idea how to calculate the problems. 
For example consider following differential equation (DE):

NDSolve[{x''[t] + 2/t x'[t] == (x[t])^2 + (x[t])^4 + 34, 
    x[∞] == 0, x'[0] == 0}, x[t], {t, 0, ∞}][[1]];
NIntegrate[(t^2)[1/2 (x'[t])^2 + (x[t])^2], {t, 0, ∞}]

It is very troubled to quantum tunnelling problem.
Please tell me how to calculate it.

bbgodfrey · Answer

This problem does not appear to have a solution.  If x[t] is to vanish at large t, then it must satisfy

x''[t] + 2/t x'[t] == 34

approximately at large t.  The solution to this equation is

First@DSolve[x''[t] + 2/t x'[t] == 34, x[t], t]
(* {x[t] -> (17 t^2)/3 - C[1]/t + C[2]} *)

For no values of the two constants does this solution approach zero as t approaches ∞.

How can I combine NIntegrate with NDSolve?

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