How can I combine NIntegrate with NDSolve?

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.

• is this different than your other question how-can-i-perform-nintegrate-with-ndsolve ? Dec 12, 2015 at 5:13
• I feel like this isn't really a problem of getting NIntegrate and NDSolve to play nice together (because really that's not too hard). Rather, it seems to me that you're not even getting a solution to your differential equation. Concentrate on that first. Dec 12, 2015 at 5:46

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]

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