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Byju's Answer
Standard XII
Mathematics
Integration Using Substitution
If ∫log 2x1...
Question
If
∫
x
log
2
1
√
e
x
−
1
d
x
=
π
6
, then the value of
x
is
A
log
2
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B
log
3
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C
log
4
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D
None of these
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Solution
The correct option is
C
log
4
Let
I
=
∫
x
log
2
1
√
e
x
−
1
d
x
Put
e
x
−
1
=
t
2
⇒
e
x
d
x
=
2
t
d
t
⇒
(
t
2
+
1
)
d
x
=
2
t
d
t
⇒
d
x
=
2
t
1
+
t
2
d
t
When
x
=
log
2
,
t
=
±
1
∴
I
=
∫
√
e
x
−
1
1
2
t
t
(
1
+
t
2
)
d
t
=
2
∫
√
e
x
−
1
1
1
1
+
t
2
d
t
=
2
|
tan
−
1
t
|
√
e
x
−
1
1
=
2
[
tan
−
1
√
e
x
−
1
−
tan
−
1
1
]
=
2
tan
−
1
√
e
x
−
1
−
π
2
But
I
=
∫
x
log
2
1
√
e
x
−
1
d
x
=
π
6
⇒
2
tan
−
1
√
e
x
−
1
−
π
2
=
π
6
⇒
2
tan
−
1
√
e
x
−
1
=
2
π
3
⇒
√
e
x
−
1
=
tan
(
π
3
)
=
√
3
⇒
e
x
−
1
=
3
⇒
e
x
=
4
∴
x
=
log
4
Suggest Corrections
0
Similar questions
Q.
Prove that
∫
x
l
o
g
2
1
e
x
−
1
d
x
=
l
o
g
3
2
when
x
=
log
4
.
Q.
The value of
2
log
2
+
log
3
log
48
−
log
4
is
Q.
Solve the equation in each of the following.
(i)
log
4
(
x
+
4
)
+
log
4
8
=
2
(ii)
log
6
(
x
+
4
)
−
log
6
(
x
−
1
)
=
1
(iii)
log
2
x
+
log
4
x
+
log
8
x
=
11
6
(iv)
log
4
(
8
log
2
x
)
=
2
(v)
log
10
5
+
log
10
(
5
x
+
1
)
=
log
10
(
x
+
5
)
+
1
(vi)
4
log
2
x
−
log
2
5
=
log
2
125
(vii)
log
3
25
+
log
3
x
=
3
log
3
5
(viii)
log
3
(
√
5
x
−
2
)
−
1
2
=
log
3
(
√
x
+
4
)
Q.
The domain of the function
f
(
x
)
=
log
2
(
log
3
(
log
4
x
)
)
is
Q.
Solve:
log
4
log
2
log
√
2
log
3
(
x
−
2006
)
=
0
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