Let
F
n
F_n
be a free group of rank
n
n
and
F
n
N
F_n^N
the quotient group of
F
n
F_n
by a subgroup
[
Γ
n
(
3
)
,
Γ
n
(
3
)
]
[
[
Γ
n
(
2
)
,
Γ
n
(
2
)
]
,
Γ
n
(
2
)
]
[\Gamma _n(3), \Gamma _n(3)][[\Gamma _n(2),\Gamma _n(2)],\Gamma _n(2)]
, where
Γ
n
(
k
)
\Gamma _n(k)
denotes the
k
k
-th subgroup of the lower central series of the free group
F
n
F_n
. In this paper, we determine the group structure of the graded quotients of the lower central series of the group
F
n
N
F_n^N
by using a generalized Chen’s integration in free groups. Then we apply it to the study of the Johnson homomorphisms of the automorphism group of
F
n
F_n
. In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map
F
n
→
F
n
N
F_n \rightarrow F_n^N
, we see that there appear only two irreducible components, the Morita obstruction
S
k
H
Q
S^k H_{\mathbf {Q}}
and the Schur-Weyl module of type
H
Q
[
k
−
2
,
1
2
]
H_{\mathbf {Q}}^{[k-2, 1^2]}
, in the cokernel of the rational Johnson homomorphism
τ
k
,
Q
′
=
τ
k
′
⊗
i
d
Q
\tau _{k, \mathbf {Q}}’=\tau _k’ \otimes \mathrm {id}_{\mathbf {Q}}
for
k
≥
5
k \geq 5
and
n
≥
k
+
2
n \geq k+2
.