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AIEEE 2006 Solved Paper
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© examsnet.com
Question : 136
Total: 143
The value of
∫
1
a
[
x
]
f
′
(
x
)
dx
,
a
>
1
where
[
x
]
denotes the greatest integer not exceeding
x
is
a
f
(
a
)
−
{
f
(
1
)
+
f
(
2
)
+
.
.
.
.
.
.
.
.
.
.
.
.
f
(
[
a
]
)
}
[
a
]
f
(
a
)
−
{
f
(
1
)
+
f
(
2
)
+
.
.
.
.
.
.
.
.
.
.
f
(
[
a
]
)
}
[
a
]
f
(
[
a
]
)
−
{
f
(
1
)
+
f
(
2
)
+
.
.
.
.
.
.
.
.
.
.
.
.
f
(
a
)
}
a
f
(
[
a
]
)
−
{
f
(
1
)
+
f
(
2
)
+
.
.
.
.
.
.
.
.
.
.
.
.
.
f
(
a
)
}
Validate
Solution:
Let
a
=
k
+
h
where
k
is an integer such that
[
a
]
=
k
and
0
≤
h
<
1
∴
a
∫
1
[
x
]
f
′
(
x
)
dx
=
2
∫
1
1
f
′
(
x
)
dx
+
3
∫
2
2
f
′
(
x
)
dx
+
.
.
.
k
∫
k
−
1
(
k
−
1
)
dx
+
k
+
h
∫
k
k
f
′
(
x
)
dx
{
f
(
2
)
−
f
(
1
)
}
+
2
{
f
(
3
)
−
f
(
2
)
}
+
3
{
f
(
4
)
−
f
(
3
)
}
+
.
.
.
.
.
.
⋯
+
(
k
−
1
)
{
f
(
k
)
−
f
(
k
−
1
)
}
+
k
{
f
(
k
+
h
)
−
f
(
k
)
}
=
−
f
(
1
)
−
f
(
2
)
−
f
(
3
)
.
.
.
.
.
.
−
f
(
k
)
+
k
f
(
k
+
h
)
=
[
a
]
f
(
a
)
−
{
f
(
1
)
+
f
(
2
)
+
f
(
3
)
+
.
.
.
.
.
.
f
(
[
a
]
)
}
© examsnet.com
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