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Byju's Answer
Standard XII
Mathematics
Integration by Substitution
∫ x +1/√2 x -...
Question
If
∫
x
+
1
√
2
x
−
1
d
x
=
f
(
x
)
√
2
x
−
1
+
C
, where
C
is a constant of integration, then
f
(
x
)
is equal to :
A
1
3
(
x
+
1
)
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B
2
3
(
x
+
2
)
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C
2
3
(
x
−
4
)
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D
1
3
(
x
+
4
)
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Solution
The correct option is
D
1
3
(
x
+
4
)
I
=
∫
x
+
1
√
2
x
−
1
d
x
Let
2
x
−
1
=
t
2
⇒
d
x
=
t
d
t
I
=
∫
t
2
+
1
2
+
1
t
⋅
t
d
t
=
1
2
∫
(
t
2
+
3
)
d
t
=
1
2
[
t
3
3
+
3
t
]
+
C
Put
t
=
(
2
x
−
1
)
1
/
2
I
=
1
2
(
2
x
−
1
)
1
/
2
[
2
x
−
1
3
+
3
]
+
C
=
1
2
(
2
x
−
1
)
1
/
2
[
2
x
+
8
3
]
+
C
Hence,
f
(
x
)
=
x
+
4
3
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26
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