It is shown that as
m
m
tends to infinity, the error in the integration of the Chebyshev polynomial of the first kind,
T
(
4
m
+
2
)
j
±
2
l
(
x
)
{T_{(4m + 2)j \pm 2l}}(x)
, by an
m
m
-point Gauss integration rule approaches
(
−
1
)
j
⋅
2
/
(
4
l
2
−
1
)
,
l
=
0
,
1
,
⋯
,
m
−
1
{( - 1)^j} \cdot 2/(4{l^2} - 1),l = 0,1, \cdots ,m - 1
, and
(
−
1
)
j
⋅
π
/
2
,
l
=
m
{( - 1)^j} \cdot \pi /2,l = m
, for all
j
j
.