integration of exp(x^2)*(x^2) ? i.e integration of { exp(x square) multiplying (x square) } ?

Question

by substitution method x^2=t and "integratioan byparts" integration is going to infinite.........

Answer 1

The integral cannot be evaluated in closed form. I don't see the value in the substitution
t = x². You will end up with

½ ∫ (√t)e^t dt

which is no more tractable than the original form.

Integration by parts of the original:

dv = x e^(x²), u = x
v = ½ e^(x²), du = dx

∫ x² e^(x²) dx = ½ x e^(x²) - ½ ∫ e^(x²) dx

and there is nothing more to be done with the remaining integral.