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Byju's Answer
Standard XII
Mathematics
Greatest Binomial Coefficients
If In=∫01dx/...
Question
If
I
n
=
∫
1
0
d
x
(
1
+
x
2
)
n
;
n
ϵ
N
,
then which of the following statements hold good ?
A
2
n
l
n
+
1
=
2
−
n
+
(
2
n
−
1
)
l
n
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B
l
2
=
π
8
+
1
4
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C
l
2
=
π
8
−
1
4
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D
l
3
=
π
16
+
5
48
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Solution
The correct options are
B
2
n
l
n
+
1
=
2
−
n
+
(
2
n
−
1
)
l
n
C
l
2
=
π
8
+
1
4
l
n
=
∫
1
0
(
1
+
x
2
)
−
n
d
x
l
n
=
[
(
1
+
x
2
)
−
n
⋅
x
]
1
0
+
n
∫
1
0
2
x
2
(
1
+
x
2
)
n
+
1
d
x
=
2
−
n
+
2
n
∫
1
0
(
x
2
+
1
)
−
1
(
x
2
+
1
)
n
+
1
d
x
l
n
=
2
−
n
+
2
n
[
l
n
−
l
n
+
1
]
2
n
l
n
+
1
=
2
−
n
+
(
2
n
−
1
)
l
n
now substitute
n
=
1
2
l
2
=
2
−
1
+
l
1
=
1
2
+
∫
1
0
1
1
+
x
2
d
x
=
1
2
+
(
tan
−
1
x
)
1
0
l
2
=
1
4
+
π
8
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0
Similar questions
Q.
If
I
n
=
1
∫
0
d
x
(
1
+
x
2
)
n
, where
n
∈
N
, then which of the following is/are correct ?
Q.
In the expansion of
(
x
cos
θ
+
1
x
sin
θ
)
16
, if
l
1
is the least value of the term independent of
x
when
π
8
≤
θ
≤
π
4
and
l
2
is the least value of the term independent of
x
when
π
16
≤
θ
≤
π
8
,
then the ratio
l
2
:
l
1
is equal to :
Q.
If
I
n
=
∫
1
0
d
x
(
1
+
x
2
)
n
;
n
∈
N
, then which of the following statements hold good?
Q.
If
I
n
=
∫
1
0
d
x
(
1
+
x
2
)
n
;
n
ϵ
N
,
then which of the following statements is correct ?
Q.
If
I
n
=
∫
1
0
d
x
(
1
+
x
2
)
n
:
n
∈
N
, then which of the following statements hold good?
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