Let $B_t:=B(t)$. By the Itô formula 
$$f(B_1)-f(B_0)=\int_0^1 f'(B_t)\,dB_t+\frac12\,\int_0^1 f''(B_t)\,dt$$
with $f(b):=\int_0^b e^{a^2}\,da$ (with $\int_0^b:=-\int_b^0$ for $b<0$), we have 
$$X_1=\int_0^1 f'(B_t)\,dB_t
=f(B_1)-\frac12\,\int_0^1 f''(B_t)\,dt
=f(B_1)-\int_0^1 B_t e^{B_t^2}\,dt.$$
From here, it should be not too hard to see that $EX_1$ does not exist.