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JEE Main 29 June 2022 Shift 1 Solved Paper
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© examsnet.com
Question : 17
Total: 90
∫
0
5
cos
(
π
(
x
−
[
x
2
]
)
)
d
x
where [t] denotes greatest integer less than or equal to
t
, is equal to:
[29-Jun-2022-Shift-1]
−
3
−
2
2
0
Validate
Solution:
👈: Video Solution
We know,
[
x
2
]
is discontinuous at
1
,
2
,
3
,
4
.
.
.
.
.
.
.
∴
[
x
]
is discontinuous at
2
,
4
,
6
,
8
.
.
.
.
.
In between 0 to 5 it is discontinuous at 2 and 4.
Break the integration into 3 parts
(1) 0 to 2
(2) 2 to 4
(3) 4 to 5
∴
5
∫
0
cos
(
π
(
x
−
[
x
2
]
)
)
dx
=
2
∫
0
cos
(
π
(
x
−
0
)
)
dx
+
4
∫
2
cos
(
π
(
x
−
1
)
)
dx
+
5
∫
4
cos
(
π
(
x
−
2
)
)
dx
=
2
∫
0
cos
π
x
dx
+
4
∫
2
cos
(
π
x
−
π
)
dx
+
5
∫
4
cos
(
π
x
−
2
π
)
dx
=
2
∫
0
cos
π
dx
−
4
∫
2
cos
π
x
dx
+
5
∫
4
cos
π
x
dx
=
[
s
i
n
π
x
π
]
0
2
−
[
s
i
n
π
x
π
]
2
4
+
[
s
i
n
π
x
π
]
4
5
=
0
−
0
+
0
=
0
© examsnet.com
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