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CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 9 - MCQExams.com
CBSE
Class 12 Commerce Applied Mathematics
Definite Integrals
Quiz 9
Let
I
1
=
∫
2
1
1
√
1
+
x
2
d
x
and
I
2
=
∫
2
1
1
x
d
x
. Then
Report Question
0%
I
1
>
I
2
0%
I
2
>
I
1
0%
I
1
=
I
2
0%
I
1
>
2
I
2
Explanation
I
1
=
∫
2
1
1
√
1
+
x
2
d
x
=
(
log
(
x
+
√
1
+
x
2
)
)
2
1
=
log
(
2
+
√
5
)
−
log
(
1
+
√
2
)
=
log
(
(
2
+
√
5
)
(
1
+
√
2
)
)
I
2
=
∫
2
1
1
x
d
x
=
(
log
x
)
2
1
=
log
2
−
log
1
=
log
2
∴
I
2
>
I
1
The value (s) of
∫
1
0
x
4
(
1
−
x
)
4
1
+
x
2
d
x
is (are)
Report Question
0%
22
7
−
π
0%
2
105
0%
0
0%
71
15
−
3
π
2
Explanation
Let
I
=
∫
1
0
x
4
(
1
−
x
)
4
1
+
x
2
d
x
=
∫
1
0
(
x
4
−
1
)
(
1
−
x
)
4
+
(
1
−
x
)
4
(
1
+
x
2
)
d
x
=
∫
1
0
(
x
2
−
1
)
(
1
−
x
)
4
d
x
+
∫
1
0
(
1
+
x
2
−
2
x
)
2
(
1
+
x
2
)
d
x
=
∫
1
0
{
(
x
2
−
1
)
(
1
−
x
)
4
+
(
1
+
x
2
)
−
4
x
+
4
x
2
(
1
+
x
2
)
}
d
x
=
∫
1
0
(
(
x
2
−
1
)
(
1
−
x
)
4
+
(
1
+
x
2
)
−
4
x
+
4
4
1
−
x
2
)
d
x
=
∫
1
0
(
x
6
−
4
x
5
+
5
x
4
−
4
x
2
+
4
−
4
1
+
x
2
)
d
x
=
[
x
7
7
−
4
x
6
6
+
5
x
5
5
−
4
x
3
3
+
4
x
−
4
tan
−
1
x
]
1
0
=
1
7
−
4
6
+
5
5
−
4
3
+
4
−
4
(
π
4
−
0
)
=
22
7
−
π
If
I
=
∫
1
0
x
d
x
8
+
x
3
then the smallest interval in which
I
lies is
Report Question
0%
(
0
,
1
8
)
0%
(
0
,
1
9
)
0%
(
0
,
1
10
)
0%
(
0
,
1
7
)
Explanation
I
=
∫
1
0
x
8
+
x
3
d
x
=
∫
1
0
x
(
x
+
2
)
(
x
2
−
2
x
+
4
)
d
x
=
∫
1
0
(
x
+
2
6
(
x
2
−
2
x
+
4
)
−
1
6
(
x
+
2
)
)
d
x
=
1
6
∫
1
0
(
2
x
−
2
2
(
x
2
−
2
x
+
4
)
+
3
6
x
2
−
2
x
+
4
)
d
x
−
1
6
∫
1
x
+
2
d
x
=
1
2
∫
1
0
2
x
−
2
x
2
−
2
x
+
4
d
x
+
1
2
∫
1
0
1
x
2
−
2
x
+
4
−
1
6
∫
1
0
1
x
+
2
d
x
=
[
1
12
log
(
x
2
−
2
x
+
4
)
−
1
6
log
(
x
+
2
)
]
+
tan
−
1
(
x
−
1
√
3
)
2
√
3
=
1
12
log
(
3
)
−
1
6
log
(
3
)
+
tan
−
1
(
2
√
3
)
2
√
3
−
1
12
log
(
4
)
−
1
6
log
(
2
)
+
tan
−
1
(
−
1
√
3
)
2
√
3
And this lies between
(
0
,
1
9
)
If
I
n
=
∫
π
4
0
tan
n
x
d
x
then
1
I
2
+
I
4
,
1
I
3
+
I
5
,
1
I
4
+
I
6
are in?
Report Question
0%
A
.
P
0%
H
.
P
0%
G
.
P
0%
None of these
Explanation
I
n
=
∫
π
/
4
0
tan
n
−
2
x
tan
2
x
d
x
=
∫
π
/
4
0
tan
n
−
2
x
(
sec
2
x
−
1
)
d
x
=
∫
π
/
4
0
tan
n
−
2
x
sec
2
x
d
x
−
∫
π
/
4
0
tan
n
−
2
x
d
x
I
n
=
[
tan
n
−
1
x
n
−
1
]
π
/
4
0
−
I
n
−
2
∴
I
n
+
I
n
−
2
=
1
n
−
1
Substitute
n
=
4
,
5
,
6
,
.
.
.
.
we get
I
4
+
I
2
,
I
5
+
I
3
,
I
6
+
I
4
,
.
.
.
.
are respectively
1
3
,
1
4
,
1
5
⋯
which are in H.P. and hence their reciprocals are in A.P.
If
I
t
=
∫
π
2
0
sin
2
t
x
sin
2
x
d
x
then ,
I
1
,
I
2
,
I
3
are in
Report Question
0%
A.P.
0%
H.P.
0%
G.P.
0%
None of these
Explanation
Consider
I
t
+
2
+
I
t
−
2
I
t
+
1
∫
π
/
2
0
(
sin
2
t
x
+
sin
2
(
t
+
2
)
x
−
2
sin
2
(
t
+
1
)
x
)
d
x
sin
2
x
∫
π
/
2
0
(
sin
2
(
t
+
2
)
x
−
sin
2
(
t
+
1
)
x
−
sin
2
(
t
+
1
)
x
+
sin
2
t
x
)
d
x
sin
2
x
∫
π
/
2
0
[
sin
(
2
t
+
3
)
x
−
sin
(
2
t
+
1
)
x
]
d
x
sin
x
=
2
∫
π
/
2
0
cos
(
2
t
+
2
)
x
d
x
=
2
(
sin
(
2
t
+
2
)
x
2
(
t
+
1
)
)
π
/
2
0
=
1
(
t
+
1
)
(
0
)
=
0
∴
I
t
,
I
t
+
1
,
I
t
+
2
ϵ
A
.
P
.
Short Cut Method :
Substitute
t
=
1
,
2
,
3
in given integral we get
I
1
=
π
2
,
I
2
=
π
,
I
3
=
3
π
2
I
1
,
I
2
,
I
3
ϵ
A
.
P
.
The value of
∫
π
/
2
0
sin
θ
log
(
sin
θ
)
d
θ
equals
Report Question
0%
log
e
(
1
e
)
0%
log
2
e
0%
log
e
2
−
1
0%
log
e
(
e
2
)
Explanation
Let
I
=
∫
π
2
0
sin
θ
log
(
sin
θ
)
d
θ
=
1
2
∫
π
2
0
sin
θ
log
(
sin
2
θ
)
d
θ
(
∵
log
e
a
m
=
m
log
e
a
)
=
1
2
∫
π
2
0
sin
θ
log
(
1
−
cos
2
θ
)
d
θ
Take
t
=
cos
θ
⇒
sin
θ
d
θ
=
−
d
t
θ
=
0
,
t
=
1
and
θ
=
π
2
,
t
=
0
Then,
I
=
1
2
∫
1
0
log
(
1
−
t
2
)
d
t
=
−
1
2
[
∫
1
0
(
t
2
+
t
4
2
+
t
6
3
+
.
.
.
∞
)
d
t
]
=
−
1
2
[
1
3
+
1
2
⋅
5
+
1
7
⋅
3
+
.
.
.
∞
]
=
−
[
1
2.3
+
1
4
⋅
5
+
1
6
⋅
7
+
.
.
.
.
∞
]
=
−
[
(
1
2
−
1
3
)
+
(
1
4
−
1
5
)
+
(
1
6
−
1
7
)
+
.
.
.
.
∞
]
=
−
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+
1
7
+
.
.
.
.
∞
=
(
1
−
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+
1
7
+
.
.
.
.
∞
)
−
1
=
log
e
2
−
1
=
log
e
2
−
log
e
=
log
e
(
2
e
)
Ans: C
If
f
(
x
)
=
x
∫
1
tan
−
1
t
t
d
t
∀
∈
R
, then the value of
f
(
e
2
)
−
f
(
1
e
2
)
is
Report Question
0%
0
0%
π
2
0%
π
0%
2
π
The value(s) of
∫
1
0
x
4
(
1
−
x
)
4
1
+
x
2
d
x
is (are)
Report Question
0%
22
7
−
π
0%
2
105
0%
0
0%
71
15
−
3
π
2
If
2
f
(
x
)
+
f
(
−
x
)
=
1
x
sin
(
x
−
1
x
)
, then the value of
∫
e
1
e
f
(
x
)
d
x
, is
Report Question
0%
1
0%
0
0%
e
0%
−
1
Explanation
Since,
2
f
(
x
)
+
f
(
−
x
)
=
1
x
sin
(
x
−
1
x
)
...(i)
∴
2
f
(
−
x
)
+
f
(
x
)
=
1
x
sin
(
x
−
1
x
)
(replace
x
by
−
x
) ...(ii)
⟹
f
(
x
)
=
f
(
−
x
)
[subtracting equations (i) and (ii)]
∴
3
f
(
x
)
=
1
x
sin
(
x
−
1
x
)
Hence,
I
=
∫
e
1
e
f
(
x
)
d
x
=
1
3
∫
e
1
e
1
x
sin
(
x
−
1
x
)
d
x
Now, put
x
=
1
t
⇒
d
x
=
−
1
t
2
d
t
∴
I
=
1
3
∫
1
e
e
t
sin
(
1
t
−
t
)
.
(
−
1
t
2
)
d
t
=
1
3
∫
1
e
e
1
t
sin
(
t
−
1
t
)
d
t
=
−
1
3
∫
e
1
e
1
t
sin
(
t
−
1
t
)
d
t
⇒
I
=
−
I
⇒
2
I
=
0
⇒
I
=
0
∴
∫
e
1
e
f
(
x
)
d
x
=
0
.
If
I
=
∫
π
1
/
π
1
x
⋅
sin
(
x
−
1
x
)
d
x
then I is equal to
Report Question
0%
0
0%
π
0%
π
−
1
π
0%
π
+
1
π
Explanation
Let
I
=
∫
π
1
π
1
x
⋅
sin
(
x
−
1
x
)
d
x
Substitute
x
=
1
t
∴
I
=
∫
1
π
π
t
sin
(
1
t
−
t
)
(
−
1
t
2
)
d
t
=
−
∫
π
1
π
1
t
sin
(
t
−
1
t
)
d
t
−
I
⇒
2
I
=
0
⇒
I
=
0
If
x
satisfies the equation
(
∫
1
0
d
t
t
2
+
2
t
cos
α
+
1
)
x
2
−
(
∫
3
−
3
t
2
sin
2
t
t
2
+
1
d
t
)
x
−
2
=
0
for
(
0
<
α
<
π
)
then the value of
x
is?
Report Question
0%
±
√
α
2
sin
α
0%
±
√
2
sin
α
α
0%
±
√
α
sin
α
0%
±
2
√
sin
α
α
Explanation
(
∫
1
0
d
t
t
2
+
2
t
cos
α
+
1
)
x
2
−
(
∫
3
−
3
t
2
sin
2
t
t
2
+
1
d
t
)
x
−
2
=
0
Here,
∫
3
−
3
t
2
sin
2
t
t
2
+
1
d
t
=
0
(
∵
f
(
t
)
=
t
2
sin
2
t
t
2
+
1
is odd
)
So, the equation becomes
(
∫
1
0
d
t
t
2
+
2
t
cos
α
+
1
)
x
2
−
2
=
0
....(1)
Now,
∫
1
0
d
t
t
2
+
2
t
cos
α
+
1
=
∫
1
0
d
t
(
t
+
cos
α
)
2
+
sin
2
α
=
1
sin
α
[
tan
−
1
(
t
+
cos
α
sin
α
)
]
1
0
=
1
sin
α
[
tan
−
1
cot
α
2
−
tan
−
1
cot
α
]
=
1
sin
α
[
π
2
−
α
2
−
π
2
+
α
]
=
α
2
sin
α
So, equation (1) becomes
α
2
sin
α
x
2
−
2
=
0
⇒
x
=
±
2
√
sin
α
α
Hence, option D.
The tangent to the graph of the function
y
=
f
(
x
)
at the point with abscissa x = a forms with the x-axis an angle of
π
/
3
and at the point with abscissa x = b at an angle of
π
/
4
, then the value of the integral,
∫
b
a
f
′
(
x
)
.
f
″
(
x
)
d
x
is equal to
Report Question
0%
1
0%
0
0%
−
√
3
0%
-1
Explanation
Given , at
x
=
a
,
d
y
d
x
=
tan
π
3
=
√
3
⇒
f
′
(
a
)
=
√
3
Also, at
x
=
b
,
d
y
d
x
=
tan
π
4
=
1
⇒
f
′
(
b
)
=
1
Now,
∫
b
a
f
′
(
x
)
.
f
″
(
x
)
d
x
Put
f
′
(
x
)
=
t
⇒
f
″
(
x
)
d
x
=
d
t
=
∫
1
√
3
t
d
t
=
−
∫
√
3
1
t
d
t
=
−
[
t
2
2
]
√
3
1
=
−
1
Let
F
(
x
)
=
f
(
x
)
+
f
(
1
x
)
where
f
(
x
)
=
∫
x
1
log
t
1
+
t
d
t
Then
F
(
e
)
is equal to?
Report Question
0%
1
0%
2
0%
1
/
2
0%
0
Explanation
F
(
x
)
=
∫
x
1
ln
t
1
+
t
d
t
+
∫
1
/
x
1
ln
t
1
+
t
d
t
F
(
x
)
=
∫
x
1
(
ln
t
1
+
t
+
ln
t
(
1
+
t
)
t
)
d
t
=
∫
t
1
ln
t
t
d
t
=
1
2
(
ln
x
)
2
F
(
e
)
=
1
/
2
The value of
∫
π
/
2
0
d
θ
5
+
3
cos
θ
is?
Report Question
0%
tan
−
1
1
2
0%
tan
−
1
1
3
0%
1
2
tan
−
1
1
2
0%
1
3
tan
−
1
1
3
Explanation
Let
I
=
∫
π
/
2
0
d
θ
5
+
3
cos
θ
=
∫
π
/
2
0
d
θ
5
+
3.
1
−
tan
2
θ
2
1
+
tan
2
θ
2
=
∫
π
/
2
0
sec
2
θ
2
d
θ
8
+
2
tan
2
θ
2
=
∫
π
/
2
0
1
2
sec
2
(
θ
2
)
d
θ
4
+
tan
2
(
θ
2
)
=
∫
1
0
d
t
4
+
t
2
[
Put
tan
θ
2
=
t
So,
1
2
sec
2
θ
2
d
θ
=
d
t
]
∴
I
=
=
1
2
[
tan
−
1
t
2
]
1
0
=
1
2
(
tan
−
1
1
2
−
tan
−
1
0
)
=
1
2
tan
−
1
1
2
Evaluate :
∞
∫
0
d
x
(
1
+
x
2
)
4
Report Question
0%
π
32
0%
3
π
32
0%
5
π
32
0%
7
π
32
Let
f
:
R
→
R
+
and
I
I
=
∫
k
1
−
k
x
f
(
x
(
1
−
x
)
)
d
x
,
I
2
=
∫
k
1
−
k
f
(
x
(
1
−
x
)
)
d
x
where
2
k
−
1
>
0
. Then
I
I
I
2
is
Report Question
0%
2
0%
k
0%
1/2
0%
1
The value of
∫
∞
0
log
x
a
2
+
x
2
Report Question
0%
2
π
log
a
a
0%
π
log
a
2
a
0%
π
log
a
0%
0
∫
1
−
1
cot
−
1
(
x
+
x
3
1
+
x
4
)
d
x
is equal to
Report Question
0%
2
π
0%
π
2
0%
0
0%
π
∫
π
/
2
0
sin
x
log
(
sin
x
)
d
x
=
Report Question
0%
log
e
e
0%
log
e
2
0%
log
e
(
e
/
2
)
0%
log
e
(
2
/
e
)
Explanation
I
=
∫
π
/
2
0
sin
x
log
(
sin
x
)
d
x
=
1
2
∫
π
/
2
0
sin
x
log
sin
2
x
d
x
=
1
2
∫
π
/
2
0
sin
x
log
(
1
−
cos
2
x
)
d
x
=
1
2
∫
1
0
log
(
1
−
t
2
)
d
t
where
t
=
cos
x
=
1
2
∫
1
0
[
−
t
2
−
(
t
2
)
2
2
−
(
t
2
)
3
3
−
⋯
]
d
t
=
−
[
1
2.3
+
1
4.5
+
1
6.7
+
⋯
]
=
−
[
(
1
2
−
1
3
)
+
(
1
4
−
1
5
)
+
(
1
6
−
1
7
)
+
⋯
]
=
−
1
+
(
1
−
1
2
+
1
3
−
1
4
+
1
5
−
⋯
)
.
=
−
log
e
e
+
log
e
(
1
+
1
)
=
log
e
(
2
/
e
)
∫
∞
0
1
1
+
x
n
d
x
,
∀
n
>
1
is equal to?
Report Question
0%
2
∫
∞
0
1
1
+
x
d
x
0%
∫
∞
−
∞
1
1
+
x
n
d
x
0%
∫
∞
1
d
x
(
x
n
−
1
)
1
/
n
0%
∫
1
0
1
(
1
−
x
n
)
1
/
n
Explanation
Let
I
=
∫
∞
0
1
1
+
x
n
d
x
=
∫
0
∞
−
d
x
1
+
x
n
Substitute
x
=
1
y
⇒
d
x
=
−
1
y
2
I
=
∫
∞
0
1
y
2
d
y
1
+
(
1
y
)
n
=
∫
∞
0
y
n
−
1
y
(
1
+
y
n
)
d
y
Substitute
t
n
−
1
+
y
n
⇒
n
t
n
−
1
d
t
=
n
y
n
−
1
d
y
I
=
∫
∞
1
d
t
t
(
t
n
−
t
)
1
/
n
=
∫
1
∞
−
1
t
2
.
d
t
(
1
−
1
t
n
)
1
/
n
Substitute
z
=
1
t
⇒
d
x
=
−
1
t
2
d
t
I
=
∫
1
0
d
z
(
1
−
z
n
)
1
/
n
=
∫
1
0
d
x
(
1
−
x
n
)
1
/
n
If
I
1
=
∫
π
/
2
0
x
sin
x
d
x
and
I
2
=
∫
π
/
2
0
tan
−
1
x
x
d
x
,
then
I
1
I
2
=
Report Question
0%
1
2
0%
1
0%
2
0%
π
2
Explanation
Substitute
x
=
tan
θ
∴
I
2
=
∫
π
/
4
0
θ
tan
θ
.
sec
2
θ
d
θ
or
I
2
=
∫
π
/
4
0
2
θ
2
sin
θ
cos
θ
d
θ
=
∫
π
/
4
0
2
θ
sin
2
θ
d
θ
Now substitute
2
θ
=
t
∴
I
2
=
1
2
∫
π
/
2
0
t
sin
t
d
t
=
1
2
I
1
∴
I
1
I
2
=
2
⇒
(
C
)
I
f
∫
1
−
1
g
(
x
)
1
+
t
2
d
t
=
f
(
x
)
,
w
h
e
r
e
,
g
(
x
)
=
sin
x
, then
f
′
(
π
3
)
equals
Report Question
0%
π
4
0%
does not exist
0%
π
√
3
4
0%
None of these
Explanation
f
(
x
)
=
∫
1
−
1
sin
x
1
+
t
2
d
t
=
sin
x
∫
1
−
1
1
1
+
t
2
d
t
=
sin
x
[
tan
−
1
t
]
1
−
1
=
sin
x
[
π
4
+
π
4
]
=
sin
x
[
π
2
]
∴
f
′
(
x
)
=
π
2
cos
x
⇒
f
′
(
π
3
)
=
π
2
1
2
=
π
4
The value of
∫
1
/
e
1
f
(
x
)
d
x
+
∫
e
1
f
(
x
)
d
x
where
f
(
x
)
is given as
l
o
g
x
1
+
x
equals
Report Question
0%
0
0%
−
1
2
0%
1
2
0%
1
Explanation
Let
l
1
=
∫
1
/
e
1
l
o
g
x
1
+
x
d
x
and
l
2
=
∫
0
1
l
o
g
x
1
+
x
d
x
Now for
l
1
=
∫
1
/
e
1
l
o
g
t
1
+
x
Substitute
x
=
1
t
⇒
d
x
=
−
1
t
2
l
1
=
∫
e
1
l
o
g
t
t
(
1
+
t
)
d
t
=
∫
e
1
l
o
g
x
x
(
1
+
x
)
d
x
( Replacing
t
→
x
)
∴
l
1
+
l
2
=
∫
e
1
l
o
g
x
x
(
1
+
x
)
d
x
+
∫
e
1
l
o
g
x
1
+
x
d
x
=
∫
e
1
(
l
o
g
(
x
)
)
(
1
+
x
)
x
(
1
+
x
)
d
x
=
∫
e
1
l
o
g
x
x
d
x
=
1
2
[
(
l
o
g
x
)
]
e
1
=
1
2
[
l
o
g
e
e
−
l
o
g
e
1
]
=
1
2
The value of
∫
tan
x
1
/
e
t
1
+
t
2
d
t
+
∫
cot
x
1
/
e
d
t
t
(
1
+
t
2
)
is
Report Question
0%
1
/
2
0%
1
0%
π
/
4
0%
none of these
Explanation
∫
tan
x
1
/
e
t
1
+
t
2
d
t
+
∫
cot
x
1
/
e
d
t
t
(
1
+
t
2
)
=
∫
tan
x
1
/
e
t
1
+
t
2
d
t
+
∫
cot
x
1
/
e
(
1
t
−
1
1
+
t
2
)
d
t
=
1
2
[
log
(
1
+
t
2
)
]
tan
x
1
/
e
+
[
log
t
]
cot
x
1
/
e
−
[
tan
−
1
t
]
cot
x
1
/
e
=
1
Evaluate :
∫
1
√
2
−
1
√
2
x
8
1
−
x
4
×
[
sin
−
1
(
1
−
2
x
2
)
+
cos
−
1
(
2
x
√
1
−
x
2
)
]
d
x
Report Question
0%
π
[
1
2
log
√
2
+
1
√
2
−
1
+
tan
−
1
1
√
2
−
21
10
√
2
]
0%
π
[
1
2
log
√
2
+
1
√
2
−
1
+
tan
−
1
1
√
2
+
21
10
√
2
]
0%
π
[
1
2
log
√
2
−
1
√
2
−
1
+
tan
−
1
1
√
2
−
21
10
√
2
]
0%
π
[
1
2
log
√
2
−
1
√
2
−
1
+
tan
−
1
1
√
2
+
21
10
√
2
]
Explanation
In
N
r
put
x
=
sin
θ
,
then
sin
−
1
(
1
−
2
sin
2
θ
)
+
cos
−
1
(
2
sin
θ
cos
θ
)
=
sin
−
1
(
cos
2
θ
)
+
cos
−
1
(
sin
2
θ
)
=
π
2
−
cos
−
1
(
cos
2
θ
)
+
π
2
−
sin
−
1
(
sin
2
θ
)
=
π
−
2
θ
−
2
θ
=
π
−
4
sin
−
1
x
∴
I
=
∫
1
√
2
−
1
√
2
x
8
1
−
x
4
[
π
−
4
sin
−
∣
x
]
d
x
=
2
∫
1
√
2
0
π
x
8
1
−
x
4
d
x
+
0
by Prop.V
=
2
π
∫
1
√
2
0
x
8
−
1
+
1
1
−
x
4
d
x
=
2
π
∫
1
√
2
0
[
−
(
x
4
+
1
)
+
1
(
1
−
x
2
)
(
1
+
x
2
)
]
d
x
=
2
π
∫
1
√
2
0
−
(
x
4
+
1
)
+
1
2
{
1
1
−
x
2
+
1
1
+
x
2
}
d
x
=
2
x
[
−
(
x
5
5
+
x
)
+
1
4
log
1
+
x
1
−
x
+
1
2
tan
−
1
x
]
1
/
√
2
0
=
π
[
1
2
log
√
2
+
1
√
2
−
1
+
tan
−
1
1
√
2
−
21
10
√
2
]
Ans: A
Solve :-
∫
1
0
d
x
(
x
2
+
1
)
3
/
2
=
Report Question
0%
1
0%
1
√
2
0%
1
2
0%
√
2
Solve:-
1
∫
0
d
x
(
x
2
+
1
)
3
/
2
Report Question
0%
1
0%
√
2
0%
1
/
2
0%
1
√
2
If
I
=
∫
1
/
√
3
0
d
x
(
1
+
x
2
)
√
1
−
x
2
then
I
is equal to
Report Question
0%
π
/
2
0%
π
/
2
√
2
0%
π
/
4
√
2
0%
π
/
4
Explanation
Let
I
=
∫
1
(
1
+
x
2
)
√
1
−
x
2
d
x
Substitute
x
=
sin
t
⇒
d
x
=
cos
t
d
t
I
=
∫
1
sin
2
t
+
1
d
t
Multiply numerator and denominator by
csc
2
t
I
=
∫
csc
2
t
csc
2
t
+
1
d
t
=
∫
csc
2
t
cot
2
t
+
2
d
t
Substitute
u
=
cot
t
⇒
d
u
=
−
csc
2
t
d
t
I
=
−
∫
1
u
2
+
2
d
u
=
−
1
√
2
tan
−
1
u
√
2
=
−
1
√
2
tan
−
1
cot
t
√
2
=
−
1
√
2
tan
−
1
cot
sin
−
1
x
√
2
Therefore
∫
1
/
√
3
0
1
(
1
+
x
2
)
√
1
−
x
2
d
x
=
π
4
√
2
∫
π
/
3
0
cos
θ
3
+
4
sin
θ
d
θ
=
λ
log
3
+
2
√
3
3
then
λ
equals
Report Question
0%
1
2
0%
1
3
0%
1
4
0%
1
8
Explanation
Given
∫
π
/
3
0
cos
θ
3
+
4
sin
θ
d
θ
=
λ
log
3
+
2
√
3
3
....(1)
Consider,
I
=
∫
π
/
3
0
cos
θ
3
+
4
sin
θ
d
θ
Substitute
sin
θ
=
t
⇒
cos
θ
d
θ
=
d
t
I
=
∫
√
3
/
2
0
d
t
3
+
4
t
Substitute
3
+
4
t
=
u
4
d
t
=
d
u
∴
I
=
1
4
∫
3
+
2
√
3
3
d
u
u
=
1
4
[
log
u
]
3
+
2
√
3
3
I
=
1
4
[
log
(
3
+
2
√
3
)
−
log
3
]
⇒
I
=
1
4
log
3
+
2
√
3
3
So, on comparing with (1), we get
λ
=
1
4
The value of
∫
tan
x
1
/
e
t
d
t
1
+
t
2
+
∫
cot
x
1
/
e
d
t
t
(
1
+
t
2
)
is
Report Question
0%
1
2
+
t
a
n
2
x
0%
1
0%
π
/
4
0%
2
π
∫
1
−
1
d
t
1
+
t
2
Explanation
∫
tan
x
1
/
e
t
1
+
t
2
d
t
+
∫
cot
x
1
/
e
d
t
t
(
1
+
t
2
)
=
∫
tan
x
1
/
e
t
1
+
t
2
d
t
+
∫
cot
x
1
/
e
(
1
t
−
1
1
+
t
2
)
d
t
=
1
2
[
log
(
1
+
t
2
)
]
tan
x
1
/
e
+
[
log
t
]
cot
x
1
/
e
−
[
tan
−
1
t
]
cot
x
1
/
e
=
1
If
I
n
=
∫
π
/
4
0
tan
n
x
×
sec
2
x
d
x
,
then
I
1
,
I
2
,
I
3
,
.......are in
Report Question
0%
A.P.
0%
G.P.
0%
H.P.
0%
none
Explanation
I
n
=
∫
π
/
4
0
tan
n
x
sec
2
x
d
x
Let,
t
=
tan
x
⇒
sec
2
x
d
x
=
d
t
Therefore,
I
n
=
∫
1
0
t
n
d
t
=
(
t
n
+
1
)
1
0
=
1
n
Therefore,
I
1
,
I
2
,
I
3
⋯
are
1
2
,
1
3
,
1
4
,
⋯
which are in H.P
Ans: C
The value of the integral
∫
β
α
d
x
√
(
x
−
α
)
(
β
−
x
)
for
β
>
α
, is
Report Question
0%
sin
−
1
α
/
β
0%
π
/
2
0%
sin
−
1
β
/
2
α
0%
π
Explanation
Given :
∫
β
α
d
x
√
(
x
−
α
)
(
β
−
x
)
I
=
∫
β
α
d
x
√
(
x
−
α
)
(
β
−
x
)
=
∫
β
α
d
x
√
−
α
β
+
x
(
β
+
α
)
−
x
2
I
=
∫
β
α
d
x
√
(
α
+
β
)
2
4
−
α
β
−
[
x
−
β
+
α
2
]
2
I
=
∫
β
α
d
x
√
(
β
−
α
)
2
4
−
[
x
−
(
β
+
α
)
2
]
2
we know that,
∫
1
√
a
2
−
x
2
d
x
=
sin
−
1
(
x
a
)
+
c
I
=
sin
−
1
[
x
−
(
β
+
α
2
)
(
β
−
α
2
)
]
β
α
=
sin
−
1
[
(
β
−
α
2
)
(
β
+
α
2
)
]
−
sin
−
1
[
(
α
−
β
2
)
(
β
−
α
2
)
]
I
=
sin
−
1
(
1
)
−
sin
−
1
(
−
1
)
I
=
π
2
+
π
2
=
π
Hence the correct answer is
π
∫
π
/
4
π
/
6
d
x
sin
2
x
is equal to
Report Question
0%
1
2
log
(
−
1
)
0%
log
(
−
1
)
0%
log
3
0%
1
2
log
√
3
Explanation
Let
I
=
∫
1
sin
2
x
d
x
=
∫
csc
2
x
d
x
Put
2
x
=
t
⇒
2
d
x
=
d
t
I
=
1
2
∫
csc
t
d
t
Multiply numerator and denominator by
cot
t
+
csc
t
I
=
1
2
∫
−
−
csc
2
t
−
cot
t
csc
t
cot
t
+
csc
t
d
t
Put
cot
t
+
csc
t
=
u
⇒
(
−
csc
2
t
−
cot
t
csc
t
)
d
t
=
d
u
I
=
−
1
2
∫
1
u
d
u
=
−
1
2
log
u
=
−
1
2
log
(
cot
t
+
csc
t
)
=
−
1
2
log
(
cot
2
x
+
csc
2
x
)
=
−
1
2
log
(
cot
x
)
Hence
∫
π
/
4
π
/
6
1
sin
2
x
d
x
=
−
1
2
[
log
(
cot
x
)
]
π
/
4
π
/
6
=
1
2
log
√
3
Value of
∫
1
0
d
x
(
1
+
x
2
)
√
1
−
x
2
is?
Report Question
0%
π
2
√
2
0%
π
√
2
0%
√
2
π
0%
2
√
2
π
Explanation
Let
I
=
∫
d
x
(
1
+
x
2
)
√
1
−
x
2
Substitute
x
=
1
t
⇒
d
x
=
−
1
t
2
d
t
I
=
∫
−
t
d
t
(
t
2
+
1
)
√
t
2
−
1
Substitute
t
2
−
1
=
u
2
⇒
2
t
d
t
=
2
u
d
u
I
=
−
∫
u
d
u
(
u
2
+
1
)
u
=
−
∫
1
u
2
+
(
√
2
)
2
d
u
=
−
1
√
2
tan
−
1
(
u
√
2
)
=
−
1
√
2
tan
−
1
(
√
2
x
√
1
−
x
2
)
Therefore
∫
1
0
I
d
x
=
π
2
√
2
If
3
+
2
∫
1
0
x
2
e
−
x
2
d
x
=
∫
1
0
e
−
x
2
d
x
then the value of
β
is
Report Question
0%
1
e
0%
e
0%
1
2
e
0%
2
e
If
I
n
=
∫
1
0
d
x
(
1
+
x
2
)
n
;
n
∈
N
, then which of the following statements hold good?
Report Question
0%
2
n
I
n
+
1
=
2
−
n
+
(
2
n
−
1
)
I
n
0%
I
2
=
π
8
+
1
4
0%
I
2
=
π
8
−
1
4
0%
I
3
=
π
16
−
5
48
If
∫
x
1
/
2
x
1
/
2
−
x
1
/
3
d
x
=
A
x
6
5
x
5
/
6
+
3
2
x
2
/
3
+
B
√
x
+
C
x
1
/
3
+
D
x
1
/
6
+
E
In
(
x
1
/
6
−
1
)
=
k
,
then A+B+C+D+E=
Report Question
0%
6
0%
12
0%
18
0%
17
The value of
∫
1
1
/
2
d
x
x
√
3
x
2
+
2
x
−
1
is?
Report Question
0%
π
/
2
0%
π
/
3
0%
π
/
6
0%
π
/
√
2
Explanation
Let
I
=
∫
1
x
√
3
x
2
+
2
x
−
1
d
x
=
∫
1
x
√
(
√
3
x
+
1
√
3
)
2
−
4
3
d
x
Put
t
=
√
3
x
+
1
√
3
⇒
d
t
=
√
3
d
t
I
=
1
√
3
∫
−
1
(
√
3
−
3
t
)
√
t
2
−
4
3
d
t
=
−
3
∫
1
(
√
3
−
3
t
)
√
t
2
−
4
3
d
t
I
=
3
∫
2
3
√
3
s
2
+
√
3
d
s
=
2
√
3
∫
1
3
s
2
+
1
d
s
=
2
tan
−
1
(
√
3
s
)
=
tan
−
1
(
x
−
1
√
3
x
2
+
2
x
−
1
)
∴
∫
1
1
/
2
1
x
√
3
x
2
+
2
x
−
1
d
x
=
π
6
If
I
=
∫
∞
1
x
2
−
2
x
3
√
x
2
−
1
d
x
, then
I
equals
Report Question
0%
−
1
0%
0
0%
π
/
2
0%
π
−
√
3
Explanation
I
=
∫
∞
1
x
2
−
2
x
3
√
x
2
−
1
d
x
Put Put
x
=
sec
u
⇒
d
x
=
tan
u
sec
u
d
u
I
=
∫
π
2
0
cos
2
u
(
sec
2
u
−
2
)
d
u
=
∫
π
2
0
(
1
−
sin
2
u
)
(
sec
2
u
−
2
)
d
u
=
∫
π
2
0
(
2
sin
2
u
−
tan
2
u
+
sec
2
u
−
2
)
d
u
=
−
∫
π
2
0
d
u
+
2
∫
π
2
0
sin
2
u
d
u
=
−
∫
π
2
0
d
u
−
∫
π
2
0
cos
2
u
d
u
+
∫
π
2
0
d
u
=
−
[
sin
2
u
2
]
π
2
0
=
0
If
0
<
α
<
π
/
2
then the value of
∫
α
0
d
x
1
−
cos
x
cos
α
is
Report Question
0%
π
/
α
0%
π
/
2
sin
α
0%
π
/
2
cos
α
0%
π
/
2
α
Explanation
Let
I
=
∫
α
0
d
x
1
−
cos
x
cos
α
=
∫
α
0
d
x
(
cos
2
x
2
+
sin
2
x
2
)
−
cos
α
(
cos
2
x
2
−
sin
2
x
2
)
=
∫
α
0
d
x
(
1
−
cos
α
)
cos
2
x
2
+
2
cos
2
α
2
sin
2
x
2
=
1
2
∫
α
0
sec
2
α
2
sec
2
x
2
tan
2
α
2
+
tan
2
x
2
d
x
Substitute
tan
x
2
=
t
I
=
∫
tan
α
2
0
sec
2
α
2
tan
2
α
2
+
t
2
d
t
=
sec
2
α
2
cot
α
2
[
tan
−
1
1
tan
α
2
]
tan
α
2
0
=
π
2
sin
α
Value of
∫
∞
a
d
x
x
4
√
a
2
+
x
2
is
Report Question
0%
2
+
√
2
3
a
4
0%
2
−
√
2
3
a
2
0%
2
−
√
2
3
a
4
0%
√
2
+
1
3
a
2
Explanation
Let
I
=
∫
∞
a
1
x
4
√
a
2
+
x
2
d
x
Substitute
x
=
a
tan
t
⇒
d
x
=
a
sec
2
t
d
t
I
=
a
∫
π
2
π
4
cot
3
t
csc
t
a
5
d
t
=
1
a
4
∫
π
2
π
4
cot
t
csc
t
(
csc
2
t
−
1
)
d
t
Substitute
υ
=
csc
t
⇒
d
υ
=
−
cot
t
csc
t
d
t
I
=
1
a
4
∫
1
√
2
(
υ
2
−
1
)
d
υ
=
1
a
2
[
υ
2
3
−
υ
]
1
√
2
=
2
−
√
2
3
a
4
The value of
∫
−
5
−
4
e
(
x
+
5
)
2
d
x
+
3
∫
2
/
3
1
/
3
e
9
(
x
−
2
/
3
)
2
d
x
is
Report Question
0%
2
/
5
0%
1
/
5
0%
1
/
2
0%
none of these
Explanation
Let
I
=
∫
−
5
−
4
e
(
x
+
5
)
2
d
x
+
3
∫
2
/
3
1
/
3
e
9
(
x
−
2
/
3
)
2
d
x
=
I
1
+
I
2
Where
I
1
=
∫
−
5
−
4
e
(
x
+
5
)
2
d
x
Put
x
+
5
=
t
I
1
=
−
∫
1
0
e
t
2
d
t
And
I
2
=
3
∫
2
/
3
1
/
3
e
9
(
x
−
2
/
3
)
2
d
x
Put
−
3
x
+
2
=
u
I
2
=
∫
1
0
e
u
2
d
u
∴
I
=
0
Value of
∫
16
0
x
1
/
4
1
+
x
1
/
2
d
x
is
Report Question
0%
8
3
0%
4
3
tan
−
1
2
0%
4
(
2
3
+
tan
−
1
2
)
0%
4
(
2
3
−
tan
−
1
2
)
Explanation
I
=
∫
16
0
4
√
x
1
+
√
x
d
x
Put
u
=
4
√
x
⇒
d
u
=
1
4
x
3
4
d
x
I
=
4
∫
2
0
(
u
2
+
1
u
2
+
1
−
1
)
d
u
=
4
∫
2
0
u
2
d
u
+
4
∫
2
0
1
u
2
+
1
d
u
−
4
∫
2
0
d
u
=
4
[
u
3
3
]
2
0
+
4
[
tan
−
1
u
]
2
0
−
4
[
u
]
2
0
=
4
(
2
3
+
tan
−
1
2
)
The solution of
x
of the equation
∫
x
√
2
d
t
t
√
t
2
−
1
=
π
2
is
Report Question
0%
2
√
2
0%
2
0%
π
0%
−
√
2
∫
1
0
1
a
x
+
b
d
x
i
s
Report Question
0%
zero
0%
l
o
g
e
a
+
b
b
0%
l
o
g
e
(
a
x
+
b
)
0%
1
a
l
o
g
e
(
a
+
b
b
)
∫
8
0
[
√
t
]
d
t
at equals to (where [.] greatest integer function.)
Report Question
0%
28
0%
11
0%
2
0%
8
If
π
4
<
α
<
π
2
, value of
∫
π
/
2
−
π
/
2
sin
2
x
√
1
+
sin
2
α
sin
x
is
Report Question
0%
−
4
3
tan
α
sec
α
0%
−
4
3
cot
α
c
o
s
e
c
α
0%
−
4
3
tan
α
c
o
s
e
c
α
0%
−
4
3
cot
α
sec
α
Explanation
Let
I
=
∫
π
/
2
−
π
/
2
sin
2
x
√
1
+
sin
2
α
sin
x
=
2
∫
π
/
2
−
π
/
2
sin
x
cos
x
√
1
+
sin
2
α
sin
x
d
x
Substitute
t
=
sin
x
⇒
d
t
=
cos
x
d
x
I
=
2
∫
1
−
1
t
√
t
sin
2
α
+
1
d
t
Substitute
u
=
t
sin
2
α
⇒
d
u
=
sin
2
α
d
t
I
=
∫
1
+
sin
2
α
1
−
sin
2
α
2
csc
2
2
α
u
−
1
√
u
[
4
3
u
3
2
csc
2
2
α
−
4
√
5
csc
2
2
α
]
1
+
sin
2
α
1
−
sin
2
α
=
−
4
3
cot
α
csc
α
If
I
1
=
∫
1
x
d
t
1
+
t
2
and
I
2
=
∫
1
/
x
1
d
t
1
+
t
2
for
x
>
0
, then
Report Question
0%
I
1
=
I
2
0%
I
1
>
I
2
0%
I
2
>
I
1
0%
I
2
=
(
π
/
2
)
−
tan
−
1
x
If
I
1
=
∫
∞
0
d
x
1
+
x
4
and
I
2
=
∫
∞
0
x
2
1
+
x
4
d
x
, then
Report Question
0%
I
1
=
I
2
0%
I
1
=
2
I
2
0%
2
I
1
=
I
2
0%
none of these
Explanation
I
2
=
∫
∞
0
x
2
1
+
x
4
d
x
=
∫
∞
0
1
x
2
1
+
1
x
4
d
x
Put
t
=
1
x
⟹
d
t
=
−
d
x
x
2
I
2
=
∫
0
∞
−
d
t
1
+
t
4
=
∫
∞
0
d
x
1
+
x
4
=
I
1
If
∫
∞
0
log
(
1
+
x
2
)
1
+
x
2
d
x
=
λ
∫
1
0
log
(
1
+
x
)
1
+
x
2
d
x
then
λ
equals
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0%
4
0%
π
0%
8
0%
2
π
Explanation
Let
I
1
=
∫
∞
0
log
(
1
+
x
2
)
1
+
x
2
d
x
and
I
2
=
∫
1
0
log
(
1
+
x
)
1
+
x
2
d
x
Put
x
=
tan
θ
in both integral
⇒
d
x
=
sec
2
θ
d
θ
I
1
=
∫
π
2
0
log
(
sec
2
θ
)
sec
2
θ
⋅
sec
2
θ
d
θ
=
−
∫
π
2
0
log
cos
2
θ
d
θ
.
.
.
.
.
.
(
1
)
Also using
(
∫
a
0
f
(
x
)
d
x
=
∫
a
0
f
(
a
−
x
)
d
x
)
I
1
=
−
∫
π
2
0
log
sin
2
θ
d
θ
.
.
.
.
.
.
.
.
(
2
)
Adding (1) and (2) we get
2
I
1
=
−
2
∫
π
2
0
log
(
sin
θ
cos
θ
)
d
θ
⇒
I
1
=
−
∫
π
2
0
log
sin
2
θ
2
d
θ
=
log
2
∫
π
2
0
d
θ
−
∫
π
2
0
log
(
sin
2
θ
)
d
θ
Put
2
θ
=
x
⇒
2
d
θ
=
d
x
⇒
I
1
=
π
2
log
2
−
1
2
∫
π
0
log
(
sin
θ
)
d
θ
=
π
2
log
2
−
∫
π
2
0
log
(
sin
θ
)
d
θ
=
π
2
log
2
+
I
1
2
, using (2)
∴
I
1
=
π
log
2
Now
I
2
=
∫
π
4
0
log
(
1
+
tan
θ
)
d
θ
⇒
I
2
=
∫
π
4
0
log
(
1
+
tan
(
π
4
−
θ
)
)
d
θ
=
∫
π
4
0
log
(
1
+
1
−
tan
θ
1
+
tan
θ
)
d
θ
⇒
I
2
=
∫
π
4
0
log
(
2
1
+
tan
θ
)
d
θ
=
log
2
∫
π
4
0
d
θ
−
∫
π
4
0
log
(
1
+
tan
θ
)
d
θ
⇒
I
2
=
π
4
log
2
−
I
2
⇒
I
2
=
π
8
log
2
Hence
λ
=
I
1
I
2
=
8
0:0:1
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers
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