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JEE Main Math Class 12 Integrals Part 2 Questions
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© examsnet.com
Question : 100
Total: 100
If the value of the integral
5
∫
0
x
+
[
x
]
e
x
−
[
x
]
d
x
=
α
e
−
+
β
,
where
α
,
β
∈
R
,
5
α
+
6
β
=
0
and [x] denotes the greatest integer less than or equal to x, then the value of
(
α
+
β
)
2
is equal to:
[26 Aug 2021 Shift 2]
100
25
16
36
Validate
Solution:
I
=
5
∫
0
x
+
[
x
]
e
x
−
[
x
]
d
x
=
α
e
−
+
β
,
I
=
1
∫
0
x
e
x
d
x
+
2
∫
1
x
+
1
e
x
−
1
d
x
+
3
∫
2
x
+
2
e
x
−
2
d
x
+
4
∫
3
x
+
3
e
x
−
3
d
x
+
5
∫
4
x
+
4
e
x
−
4
d
x
Let
I
=
I
1
+
I
2
+
I
3
+
I
4
+
I
5
Here,
,
I
2
=
2
∫
1
x
+
1
e
x
−
1
d
x
Put
x
=
t
+
1
⇒
d
x
=
d
t
=
1
∫
0
t
+
2
e
t
d
t
=
1
∫
0
t
e
t
d
t
+
1
∫
0
2
e
t
d
t
I
2
=
I
1
+
2
1
∫
0
e
−
t
d
t
=
I
1
+
2
(
1
−
e
−
1
)
Similarly,
I
3
=
I
1
+
4
(
1
−
e
−
1
)
I
4
=
I
1
+
6
(
1
−
e
−
1
)
I
5
=
I
1
+
8
(
1
−
e
−
1
)
I
=
I
1
+
I
2
+
I
3
+
I
4
+
I
5
=
5
I
1
+
(
2
+
4
+
6
+
8
)
(
1
−
e
−
1
)
=
5
l
1
+
20
(
1
−
e
−
1
)
I
1
=
1
∫
0
x
e
−
1
d
x
=
−
[
e
−
x
(
x
+
1
)
]
0
1
=
1
−
2
e
−
1
∴
5
l
1
+
20
(
1
−
e
−
1
)
=
5
(
1
−
2
e
−
1
)
+
20
(
1
−
e
−
1
)
=
25
−
30
e
−
1
∴
α
=
−
30
,
β
=
25
Also it satisfy
5
β
+
6
α
=
0
Now,
(
α
+
β
)
2
=
(
−
30
+
25
)
2
=
(
−
5
)
2
=
25
© examsnet.com
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