  In this article we present bosonic T-dualization in double space of the type
II superstring theory in pure spinor formulation. We use the action with
constant background fields obtained from the general case under some physically
and mathematically justified assumptions. Unlike Ref.\cite{bnbstip2}, where we
used the first order theory, in this article fermionic momenta are integrated
out. Full T-dualization in double space is represented as permutation of
initial $x^\mu$ and T-dual coordinates $y_\mu$. Demanding that T-dual
transformation law of the T-dual double coordinate ${}^\star Z^M=(y_\mu,x^\mu)$
is of the same form as for initial one $Z^M=(x^\mu,y_\mu)$, we obtain the form
of the T-dual background fields in terms of the initial ones. Advantage of
using the action with integrated fermionic momenta is that it gives all T-dual
background fields in terms of the initial ones. In the case of the first order
theory \cite{bnbstip2} T-dual R-R field strength was obtained out of double
space formalism under additional assumptions.
