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CBSE Questions for Class 12 Commerce Applied Mathematics Definite Integrals Quiz 4 - MCQExams.com
CBSE
Class 12 Commerce Applied Mathematics
Definite Integrals
Quiz 4
If
f
(
1
x
)
+
x
2
f
(
x
)
=
0
for
x
>
0
,
and
I
=
∫
x
1
/
x
f
(
z
)
d
z
,
1
2
≤
x
≤
2
then
I
is?
Report Question
0%
f
(
2
)
−
f
(
1
/
2
)
0%
f
(
1
/
2
)
−
f
(
2
)
0%
0
0%
None of these
Explanation
Substitute
z
=
1
t
I
=
∫
1
t
x
f
(
1
t
)
.
(
−
1
t
2
)
d
t
=
∫
1
x
x
(
−
t
2
.
f
(
t
)
)
(
−
1
t
2
)
=
∫
1
x
x
f
(
t
)
d
t
=
−
∫
x
1
x
f
(
t
)
d
t
=
−
I
∴
2
I
=
0
If
I
1
=
∫
−
5
−
4
e
(
x
+
5
)
2
d
x
and
I
2
=
3
∫
2
/
3
1
/
3
e
(
3
x
−
2
)
2
d
x
then
I
1
+
I
2
equals?
Report Question
0%
1
3
0%
−
1
3
0%
0
0%
None of these
Explanation
I
2
=
3
∫
2
3
1
3
e
(
3
x
−
2
)
2
d
x
x
=
−
y
−
3
3
d
x
=
−
d
y
3
⇒
I
2
=
3
∫
2
3
1
3
e
(
3
x
−
2
)
2
d
x
=
−
∫
−
5
−
4
e
(
y
+
5
)
2
d
x
=
−
I
1
⇒
I
1
+
I
2
=
0
If
f
(
a
+
x
)
=
f
(
x
)
, then
∀
a
>
0
,
n
ϵ
N
the value of
∫
n
a
0
f
(
x
)
d
x
equals ?
Report Question
0%
(
n
−
1
)
∫
a
0
f
(
x
)
d
x
0%
(
1
−
n
)
∫
a
0
f
(
x
)
d
x
0%
n
∫
a
0
f
(
x
)
d
x
0%
None of the above
Explanation
I
=
∫
n
a
0
f
(
x
)
d
x
Substitute
x
=
a
+
t
⇒
d
x
=
d
t
I
=
∫
(
n
−
1
)
a
0
f
(
a
+
t
)
d
t
=
(
n
−
1
)
∫
a
0
f
(
a
+
t
)
d
t
=
(
n
−
1
)
∫
a
0
f
(
a
+
x
)
d
x
=
(
n
−
1
)
∫
a
0
f
(
x
)
d
x
If
f
(
x
)
=
∫
1
−
1
sin
x
1
+
t
2
d
t
then
f
′
(
π
3
)
is
Report Question
0%
nonexistent
0%
π
/
4
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
∫
1
0
2
x
+
1
−
3
x
−
1
6
x
d
x
Report Question
0%
4
3
log
3
e
−
1
6
log
2
e
.
0%
−
4
3
log
3
e
+
1
6
log
2
e
.
0%
4
3
log
3
e
−
1
3
log
2
e
.
0%
−
2
3
log
3
e
+
1
6
log
2
e
.
Explanation
∫
1
0
2
x
+
1
−
3
x
−
1
6
x
d
x
=
∫
1
0
(
2.3
−
x
−
1
3
2
−
x
)
d
x
=
[
−
2
3
−
x
log
3
+
1
3
2
−
x
log
2
]
1
=
−
2
log
3
(
1
3
−
1
)
+
1
3
log
2
(
1
2
−
1
)
=
4
3
log
3
e
−
1
6
log
2
e
Hence, option 'A' is correct.
∫
2
1
(
x
+
1
x
)
3
/
2
x
2
−
1
x
2
d
x
Report Question
0%
5
2
√
(
5
2
)
+
8
5
√
2
0%
5
2
√
(
5
2
)
−
8
5
√
2
0%
√
(
5
2
)
−
8
5
√
2
0%
3
2
√
(
3
2
)
−
8
5
√
2
Explanation
Let
I
=
∫
2
1
(
x
+
1
x
)
3
/
2
x
2
−
1
x
2
d
x
Put
x
+
1
x
=
t
⇒
(
1
−
1
x
2
)
d
x
⇒
d
t
Therefore
I
=
∫
5
/
2
2
t
3
/
2
d
t
=
2
5
t
5
/
2
=
2
5
[
(
5
2
)
5
/
2
−
2
5
/
2
]
=
2
5
[
(
5
2
)
2
√
(
5
2
)
−
2
2
√
2
]
=
5
2
√
(
5
2
)
−
8
5
√
2
Hence, option 'B' is correct.
Evaluate the integrals
∫
b
a
log
x
x
d
x
Report Question
0%
1
2
log
(
a
b
)
⋅
log
b
a
0%
1
4
log
(
a
b
)
⋅
log
b
a
0%
log
(
a
b
)
⋅
log
b
a
0%
−
1
2
log
(
a
b
)
⋅
log
b
a
Explanation
Substitute
log
x
=
t
∴
(
1
/
x
)
d
x
=
d
t
and adjust the limits
∴
I
=
∫
t
d
t
=
[
1
2
t
2
]
log
b
log
a
∴
I
=
1
2
[
(
log
b
)
2
−
(
log
a
)
2
]
=
1
2
[
(
log
b
+
log
a
)
(
log
b
−
log
a
)
]
∴
I
=
1
2
log
(
a
b
)
⋅
log
b
a
.
Given
∫
2
1
e
x
2
d
x
=
a
,
the value of
∫
e
4
e
√
ln
(
x
)
d
x
is?
Report Question
0%
e
4
−
e
0%
e
4
−
a
0%
2
e
4
−
a
0%
2
e
4
−
e
−
a
Explanation
Given
∫
2
1
e
x
2
d
x
=
a
,
Let
I
=
∫
e
4
e
√
ln
(
x
)
d
x
Put
ln
(
x
)
=
t
2
⇒
1
x
d
x
=
2
t
d
t
∴
I
=
∫
2
1
e
t
2
.
2
t
2
d
t
=
(
t
e
t
2
)
2
1
−
∫
2
1
e
t
2
d
t
=
2
e
4
−
e
−
a
.
∫
π
/
2
0
d
x
sin
x
equals
Report Question
0%
0
0%
1
2
0%
1
0%
3
/
2
Explanation
Let
I
=
∫
π
2
0
1
s
i
n
x
d
x
=
∫
π
2
0
csc
x
d
x
Multiply numerator and denominator by
c
o
t
x
+
csc
x
, we get
∴
I
=
−
∫
π
2
0
−
c
o
t
x
csc
x
−
csc
2
x
d
x
c
o
t
x
+
csc
x
d
x
Substitute
t
=
c
o
t
x
+
csc
x
⇒
d
t
=
(
−
csc
2
x
−
c
o
t
x
csc
x
)
d
x
∴
I
=
−
∫
1
0
1
t
d
t
=
[
−
log
t
]
1
0
=
0
The value of
∫
1
0
d
x
(
x
+
1
)
√
x
2
+
2
x
is
Report Question
0%
π
/
6
0%
π
/
3
0%
π
/
2
0%
π
Explanation
Let
I
=
∫
1
0
1
(
x
+
1
)
√
x
2
+
2
x
d
x
=
∫
1
0
1
(
x
+
1
)
√
(
x
+
1
)
2
−
1
d
x
Substitute
u
=
x
+
1
⇒
d
u
=
d
x
∴
I
=
∫
2
1
1
u
√
u
2
−
1
d
u
Substitute
s
=
√
u
2
−
1
⇒
d
s
=
u
√
u
2
−
1
d
u
∴
I
=
∫
√
3
0
1
s
2
+
1
d
s
=
[
tan
−
1
s
]
√
3
0
=
π
3
Value of
∫
π
/
4
0
(
√
tan
x
−
√
cot
x
)
d
x
is
Report Question
0%
√
2
log
(
√
2
−
1
)
0%
√
2
log
(
√
2
+
1
)
0%
log
(
√
2
+
1
)
0%
log
(
√
2
−
1
)
Explanation
Let
I
=
∫
π
4
0
(
√
tan
x
−
√
c
o
t
x
)
d
x
=
−
∫
π
4
0
c
o
s
x
−
s
i
n
x
√
c
o
s
x
s
i
n
x
d
x
Substitute
(
s
i
n
x
+
c
o
s
x
)
=
t
⇒
2
s
i
n
x
c
o
s
x
=
t
2
−
1
∴
I
=
√
2
∫
√
2
1
d
t
√
t
2
−
1
=
[
√
2
log
(
t
+
√
t
2
−
1
)
]
√
2
0
=
√
2
log
(
√
2
−
1
)
The value of
∫
b
a
log
x
x
d
x
is
Report Question
0%
log
(
a
b
)
log
(
b
a
)
0%
1
2
log
(
a
b
)
log
(
b
a
)
0%
log
(
a
2
−
b
2
)
0%
(
a
+
b
)
log
(
a
+
b
)
Explanation
I
=
∫
b
a
log
x
x
d
x
=
[
log
x
.
log
x
]
b
a
−
∫
b
a
log
x
x
d
x
⇒
2
I
=
[
(
log
x
)
2
]
b
a
=
(
log
b
)
2
−
(
log
a
)
2
⇒
I
=
1
2
(
log
b
+
log
a
)
(
log
b
−
log
a
)
=
1
2
log
a
b
log
b
a
If
f
(
x
)
=
A
sin
(
π
x
2
)
+
B
,
f
′
(
1
2
)
=
√
2
and
∫
1
0
f
(
x
)
d
x
=
2
A
π
,
then the constants
A
and
B
are
Report Question
0%
A
=
π
2
,
B
=
π
2
0%
A
=
2
π
,
B
=
3
π
0%
A
=
0
,
B
=
−
4
π
0%
A
=
4
π
,
B
=
0
Explanation
f
(
x
)
=
A
sin
(
π
x
/
2
)
+
B
⟹
f
′
(
x
)
=
A
π
2
cos
(
π
x
/
2
)
⟹
f
′
(
1
/
2
)
=
A
π
2
cos
(
π
/
4
)
=
A
π
2
1
√
2
⟹
√
2
=
A
π
2
1
√
2
Hence,
A
=
4
π
∫
1
0
f
(
x
)
d
x
=
[
−
2
A
π
cos
(
π
x
/
2
)
+
B
x
]
1
0
⟹
2
A
π
=
2
A
π
+
B
Hence,
B
=
0
Hence, answer is option-(D).
The value of the integral
∫
π
/
4
0
sin
x
+
cos
x
3
+
sin
2
x
d
x
is
Report Question
0%
log
2
0%
log
3
0%
(
1
/
4
)
log
3
0%
(
1
/
8
)
log
3
Explanation
Let
I
=
∫
π
/
4
0
sin
x
+
cos
x
3
+
sin
2
x
d
x
=
∫
π
/
4
0
sin
x
+
cos
x
4
−
(
sin
x
−
cos
x
)
2
d
x
Substitute
sin
x
−
cos
x
=
t
I
=
∫
0
−
1
d
t
4
−
t
2
=
1
4
log
3
The value of
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
is
Report Question
0%
π
1
+
α
2
if
α
>
1
0%
π
α
2
−
1
if
α
>
1
0%
π
1
+
α
2
if
α
<
1
0%
π
α
2
−
1
if
α
<
1
Explanation
Let
I
=
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
=
∫
π
0
d
x
1
−
2
α
cos
x
+
α
2
Substitute
tan
x
2
=
t
⇒
1
2
sec
2
x
2
d
x
=
d
t
I
=
∫
∞
0
1
1
−
2
α
(
1
−
t
2
1
+
t
2
)
+
α
2
.
2
t
1
+
t
2
d
t
=
∫
∞
0
2
t
1
+
t
2
−
2
α
(
1
−
t
2
)
+
α
2
(
1
+
t
2
)
d
t
=
π
α
2
−
1
if
α
>
1
If
I
=
∫
2
1
d
x
(
x
+
1
)
√
x
2
−
1
then
I
is equal to
Report Question
0%
1
0%
1
/
2
0%
1
/
√
2
0%
1
/
√
3
Explanation
As form of integrand is
1
L
√
Q
Substitute
L
=
1
t
⇒
x
+
1
=
1
t
I
=
∫
1
3
1
2
(
−
1
t
2
)
d
t
1
t
√
(
1
t
−
1
)
2
−
1
=
∫
1
2
1
3
d
t
√
1
−
2
t
=
[
(
1
−
2
t
)
1
2
(
−
2
)
(
1
2
)
]
1
2
1
3
=
1
√
3
The value of
∫
1
0
d
x
e
x
+
e
−
x
is
Report Question
0%
tan
−
1
e
0%
tan
−
1
(
e
)
−
π
/
4
0%
tan
−
1
(
e
)
−
tan
−
1
(
1
/
e
)
0%
tan
−
1
(
1
/
e
)
+
π
/
4
Explanation
Let
I
=
∫
1
0
d
x
e
x
+
e
−
x
=
∫
1
0
e
x
d
x
e
2
x
+
1
Substitute
e
x
=
t
⇒
e
x
d
x
=
d
t
I
=
∫
e
1
t
t
2
+
1
d
t
=
[
tan
−
1
t
]
e
1
=
tan
−
1
e
−
π
4
The value of
∫
15
8
d
x
(
x
−
3
)
√
x
+
1
is
Report Question
0%
log
5
3
0%
1
2
log
5
3
0%
2
log
5
3
0%
√
2
log
(
√
2
−
1
)
Explanation
I
=
∫
15
8
d
x
(
x
−
3
)
√
x
+
1
Substitute
√
x
+
1
=
t
⇒
x
+
1
=
t
2
I
=
∫
4
3
2
t
(
t
2
−
4
)
t
d
t
=
[
2
2.2
log
(
t
−
2
t
+
2
)
]
4
3
=
1
2
[
log
1
3
−
log
1
5
]
=
1
2
log
5
3
If
b
>
a
,
and
I
=
∫
b
a
√
x
−
a
b
−
x
d
x
,
then
I
equals
Report Question
0%
π
2
(
b
−
a
)
0%
π
(
b
−
a
)
0%
π
/
2
0%
2
π
(
b
−
a
)
Explanation
Substitute
b
−
x
=
t
2
so that
I
=
∫
0
√
b
−
a
√
b
−
t
2
−
a
t
2
(
−
2
t
)
d
t
=
2
∫
c
0
√
c
2
−
t
2
d
t
, where
c
=
√
b
−
a
=
2
[
1
2
t
√
c
2
−
t
2
+
c
2
2
sin
−
1
(
t
c
)
]
c
0
=
0
+
c
2
sin
−
1
(
1
)
−
0
=
π
2
(
b
−
a
)
Value of
∫
2
a
0
x
3
/
2
√
2
a
−
x
d
x
is
Report Question
0%
3
π
a
2
2
0%
π
a
3
0%
√
2
π
a
3
0%
2
π
a
3
Explanation
Let
I
=
∫
2
a
0
x
3
2
√
2
a
−
x
d
x
Substitute
u
=
√
x
⇒
d
u
=
1
2
√
x
d
x
I
=
2
∫
√
2
a
0
u
4
√
2
a
−
u
2
d
u
Substitute
u
=
√
2
a
sin
t
⇒
d
u
=
√
2
a
cos
t
d
t
I
=
2
√
2
a
∫
π
2
0
(
2
√
2
a
3
2
sin
4
t
)
d
t
=
8
a
2
∫
π
4
0
sin
4
t
d
t
Using reduction formulae
∫
sin
m
x
d
x
=
−
cos
x
sin
m
−
1
x
m
+
m
−
1
m
∫
sin
m
−
2
x
d
x
I
=
[
−
2
a
2
sin
3
t
cos
t
]
π
2
0
+
6
a
2
∫
π
2
0
sin
2
t
d
t
=
0
+
6
a
2
∫
π
2
0
(
1
2
−
1
2
cos
2
t
)
d
t
=
[
3
a
2
t
−
3
a
2
sin
t
cos
t
]
π
2
0
=
3
π
a
2
2
Value of
∫
25
0
1
√
4
+
√
x
d
x
is
Report Question
0%
2
(
√
29
−
1
)
0%
2
(
√
29
−
5
)
0%
3
√
29
−
1
0%
none of these
Explanation
Let
I
=
∫
25
0
1
√
√
x
+
4
d
x
Substitute
u
=
√
x
⇒
d
u
=
1
2
√
x
d
x
I
=
2
∫
5
0
u
√
u
+
4
d
u
Substitute
s
=
u
+
4
⇒
d
s
=
d
u
I
=
2
∫
9
4
s
−
4
√
s
d
s
=
2
∫
9
4
(
√
s
−
4
√
s
)
d
s
=
[
4
s
3
/
2
3
]
9
4
−
[
16
√
s
]
9
4
=
36
−
32
3
−
48
−
32
=
−
164
3
∫
∞
0
f
(
x
+
1
x
)
ln
x
x
d
x
Report Question
0%
is equal to zero
0%
is equal to one
0%
is equal to
1
2
0%
can not be evaluated
Explanation
I
=
∫
∞
0
f
(
x
+
1
x
)
ln
x
x
d
x
Substitute
x
=
1
t
⇒
d
x
=
−
1
t
2
d
t
I
=
∫
0
∞
f
(
t
+
1
t
)
ln
(
1
/
t
)
1
/
t
.
−
d
t
t
2
=
∫
0
∞
f
(
t
+
1
t
)
ln
t
t
d
t
=
∫
0
∞
f
(
x
+
1
x
)
ln
x
x
d
x
=
−
I
⇒
2
I
=
0
⇒
I
=
0
If
∫
π
/
3
0
cos
3
+
4
sin
x
d
x
=
K
log
(
3
+
2
√
3
)
3
then K is
Report Question
0%
1
2
0%
1
3
0%
1
4
0%
1
8
Explanation
I
=
∫
π
/
3
0
cos
x
3
+
4
sin
x
d
x
Substituting
t
=
3
+
4
sin
x
⇒
d
t
=
4
cos
x
I
=
1
4
∫
3
+
2
√
3
3
1
t
d
t
=
[
log
t
4
]
3
+
2
√
3
3
=
log
(
3
+
2
√
3
)
−
log
3
4
=
1
4
log
(
3
+
2
√
3
3
)
⇒
K
=
1
4
Suppose that F(x) is an antiderivative of f(x)
=
sin
x
x
,
x
>
0
then
∫
3
1
sin
2
x
x
can be expressed as
Report Question
0%
F
(
6
)
−
F
(
2
)
0%
1
2
(
F
(
6
)
−
F
(
2
)
)
0%
1
2
(
F
(
3
)
−
F
(
1
)
)
0%
2
(
F
(
6
)
−
F
(
2
)
)
Explanation
F
(
x
)
=
∫
sin
x
x
d
x
Now
I
=
∫
3
1
sin
2
x
x
d
x
[
p
u
t
2
x
=
t
]
=
∫
6
2
2
2
sin
t
d
t
=
[
F
(
x
)
]
6
2
=
F
(
6
)
−
F
(
2
)
If
0
<
α
<
1
and
I
=
∫
1
−
1
d
x
√
1
−
2
α
x
+
α
2
then
I
equals
Report Question
0%
1
/
α
0%
2
/
α
0%
3
/
α
0%
none of these
Explanation
I
=
∫
1
−
1
(
1
−
2
α
x
+
α
2
)
−
1
/
2
d
x
=
2
(
1
−
2
α
x
+
α
2
)
1
/
2
−
2
α
]
1
−
1
=
−
1
α
[
(
[
(
1
−
2
α
+
α
2
)
1
/
2
−
(
1
+
2
α
+
α
2
)
1
/
2
]
=
−
1
α
[
√
(
α
−
1
)
2
−
√
(
α
+
1
)
2
]
=
−
1
α
[
(
1
−
α
)
−
(
1
+
α
)
]
…
(
∵
0
<
α
<
1
)
=
2
Ans: D
∫
2
1
/
2
1
x
sin
(
x
−
1
x
)
d
x
has the value equal to
Report Question
0%
0
0%
3
4
0%
5
4
0%
2
Explanation
I
=
∫
2
1
2
1
x
sin
(
x
−
1
x
)
d
x
Put
x
=
1
t
⇒
d
x
=
1
t
2
d
t
I
=
∫
1
2
2
t
sin
(
1
t
−
t
)
(
−
1
t
2
)
d
t
=
∫
1
2
2
1
t
sin
(
t
−
1
t
)
d
t
=
−
∫
2
1
2
1
t
sin
(
t
−
1
t
)
d
t
=
−
I
.
⇒
2
I
=
0
⇒
I
=
0
The value of the definite integral
∫
∞
1
(
e
x
+
1
+
e
3
−
x
)
−
1
d
x
is
Report Question
0%
π
4
e
2
0%
π
4
e
0%
1
e
2
(
π
2
−
tan
−
1
1
e
)
0%
π
2
e
2
Explanation
I
=
∫
∞
1
1
e
x
+
1
+
e
3
−
x
d
x
=
∫
∞
1
e
x
−
1
(
e
x
−
i
e
)
(
e
x
+
i
e
)
d
x
Substituting
e
x
=
t
⇒
e
x
d
x
=
d
t
I
=
1
e
∫
∞
e
1
(
t
−
i
e
)
(
t
+
i
e
)
d
t
=
1
e
∫
∞
e
1
(
e
−
i
t
)
(
e
+
i
t
)
d
t
=
1
e
∫
∞
e
(
i
2
(
e
t
+
i
e
2
)
−
i
2
(
e
t
−
i
e
2
)
)
d
t
=
[
i
log
(
e
t
+
i
e
2
)
2
e
2
−
i
log
(
e
t
−
i
e
2
)
2
e
2
]
∞
e
=
[
tan
−
1
(
e
x
−
1
)
e
2
]
∞
e
=
π
4
e
2
∫
3
π
10
π
5
s
i
n
x
s
i
n
x
+
c
o
s
x
d
x
is equal to
Report Question
0%
π
0%
π
2
0%
π
4
0%
π
20
Explanation
Let
I
=
∫
sin
x
sin
x
+
cos
x
d
x
Multiply numerator and denominator by
I
=
∫
sin
x
sin
x
+
cos
x
d
x
csc
3
x
I
=
∫
csc
2
x
csc
2
x
+
cot
x
csc
2
x
d
x
=
∫
csc
2
x
cot
3
x
+
cot
2
x
+
cot
x
+
1
d
x
Put
t
=
cot
x
⇒
d
t
=
−
csc
2
x
d
x
I
=
−
∫
1
t
3
+
t
2
+
t
+
1
d
t
=
−
∫
1
−
t
2
(
t
2
+
1
)
d
t
−
∫
1
2
(
t
+
1
)
d
t
=
1
2
∫
t
t
2
+
1
d
t
−
1
2
∫
1
t
2
+
1
d
t
−
1
2
∫
1
t
+
1
d
t
=
1
4
log
(
t
2
+
1
)
−
1
2
tan
−
1
t
−
1
2
log
(
t
+
1
)
=
1
2
(
x
−
log
(
sin
x
+
cos
x
)
)
Hence
∫
3
π
10
π
5
s
i
n
x
s
i
n
x
+
c
o
s
x
d
x
=
π
20
If
f
(
x
)
=
∫
x
1
ln
t
1
+
t
d
t
where
x
>
0
, then the value(s) of
x
satisfying the equation,
f
(
x
)
+
f
(
1
/
x
)
=
2
is
Report Question
0%
2
0%
e
0%
e
−
2
0%
e
2
Explanation
f
(
x
)
=
∫
x
1
ln
t
1
+
t
d
t
⇒
f
(
1
x
)
=
∫
1
/
x
1
ln
t
1
+
t
d
t
Substituting
t
=
1
u
⇒
d
t
=
(
−
1
u
2
)
d
u
Therefore
f
(
1
x
)
=
∫
x
1
ln
(
1
u
)
(
−
1
)
(
1
+
1
u
)
u
2
d
u
=
∫
x
1
ln
u
u
(
u
+
1
)
d
u
=
∫
x
1
ln
t
t
(
1
+
t
)
d
t
Now,
f
(
x
)
+
f
(
1
x
)
=
∫
x
1
ln
t
(
1
+
t
)
d
t
+
∫
x
1
ln
t
t
(
1
+
t
)
d
t
=
∫
x
1
(
1
+
t
)
ln
t
t
(
1
+
t
)
d
t
=
∫
x
1
ln
t
t
d
t
=
1
2
(
ln
x
)
2
Hence
f
(
x
)
+
f
(
1
x
)
=
2
⇒
x
=
e
±
2
Choose a function
f
(
x
)
such that it is integrable over every interval on the real line
Report Question
0%
f
(
x
)
=
[
x
]
0%
f
(
x
)
=
x
|
x
|
0%
f
(
x
)
=
[
s
i
n
x
]
0%
f
(
x
)
=
|
x
−
1
|
x
−
1
∫
∞
0
x
(
1
+
x
)
(
1
+
x
2
)
d
x
Report Question
0%
π
4
0%
π
2
0%
is sme as
∫
∞
0
d
x
(
1
+
x
)
(
1
+
x
2
)
0%
cannot be evaluated
Explanation
Let
I
=
∫
∞
0
x
d
x
(
1
+
x
)
(
1
+
x
2
)
Using partial fraction
=
∫
∞
0
(
x
+
1
2
(
1
+
x
2
)
−
1
2
(
1
+
x
)
)
d
x
=
(
lim
State true or false:
The average value of the function
f(x) = sin^2xcos^3x
on the interval
[ -\pi ,\pi ]
is 0.
Report Question
0%
True
0%
False
Explanation
\int _{ -\pi }^{ \pi }{ f\left( x \right) dx } =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \cos ^{ 3 }{ x } dx } \\ =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \left( 1-\sin ^{ 2 }{ x } \right) \cos { x } dx } \\ =\int _{ -\pi }^{ \pi }{ \sin ^{ 2 }{ x } \cos { x } dx } -\int _{ -\pi }^{ \pi }{ \sin ^{ 4 }{ x } \cos { x } dx }
\displaystyle ={ \left[ \frac { \sin ^{ 3 }{ x } }{ 3 } \right] }_{ -\pi }^{ \pi }-{ \left[ \frac { \sin ^{ 5 }{ x } }{ 3 } \right] }_{ -\pi }^{ \pi }=0
I_{1}
is equal to
Report Question
0%
\displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin^{2}\theta)(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {2}{3} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
0%
\displaystyle \frac {3}{2} \int_{0}^{\pi/2}(sin\theta)^{2/3}(cos\theta)^{-1/3}d\theta
Explanation
\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3}dx=(1-x^{2})^{3}_{0}+\frac {2}{3} \int_{0}^{1}\frac {x^{2}}{(1-x^{2})^{2/3}}dx
x=sin\theta, dx=cos\theta d\theta
\displaystyle I_{1}=\frac {2}{3} \int_{0}^{\pi/2}\frac {sin^{2}\theta cos\theta}{(cos\theta)^{4/3}}d\theta
\displaystyle I_{1}=\frac {2}{3} \int_{0}^{\pi/2}sin^{2}\theta (cos\theta)^{-1/3}d\theta
Ans:
A
Evaluate
\displaystyle \int_{0}^{\pi /2} \frac{dx}{2+\sin 2x}
Report Question
0%
\displaystyle \frac{2\pi }{{3}}
0%
\displaystyle \frac{\pi }{{3}}
0%
\displaystyle \frac{2\pi }{{5}}
0%
None of these
Explanation
\displaystyle \int _{ 0 }^{ 2\pi }{ \frac { dx }{ 2+\sin { 2x } } } =\int _{ 0 }^{ 2\pi }{ \frac { dx }{ 2+\frac { 2\tan { x } }{ 1+\tan ^{ 2 }{ x } } } } dx
\displaystyle =\frac { 1 }{ 2 } \int _{ 0 }^{ 2x }{ \frac { \sec ^{ 2 }{ x } }{ \tan ^{ 2 }{ x } +\tan { x+1 } } dx }
Put
\tan { x } =t\Rightarrow \sec ^{ 2 }{ x } dx=dt
\displaystyle I=\frac { 1 }{ 2 } \int { \frac { 1 }{ { { t }^{ 2 }+t+1 } } } dt.=\frac { 1 }{ 2 } \int { \frac { 1 }{ { \left( t+\frac { 1 }{ 2 } \right) }^{ 2 }-\frac { 3 }{ 2 } } } =\frac { 2\pi }{ 3 }
Report Question
0%
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
0%
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
0%
Assertion is correct but Reason is incorrect
0%
Both Assertion and Reason are incorrect
Explanation
Reason is true
Assertion
Put
\displaystyle x=\frac{1}{t}\Rightarrow dx=-\frac{1}{t^{2}}dt
\displaystyle l=-\int_{3}^{1/3}tcosec ^{99}\left ( \frac{1}{t}-t \right )\frac{1}{t^{2}}dt
\displaystyle l=-\int_{1/3}^{3}\frac{1}{t}cosec^{99}\left ( t-\frac{1}{t} \right )dt
\displaystyle l=-1
\Rightarrow 2l=0
\Rightarrow l=0
\displaystyle \int_{\tfrac{1}{\sqrt{3}}}^{0}\dfrac{dx}{\left ( 2x^{2}+1 \right )\sqrt{x^{2}+1}}
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\displaystyle - \tan^{-1} \dfrac{1}{2}
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\displaystyle \tan^{-1} 1
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\displaystyle - \tan^{-1} \dfrac{1}{3}
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\displaystyle \tan^{-1} \dfrac{1}{\sqrt 2}
Explanation
I=\displaystyle \int_{\frac{1}{\sqrt{3}}}^{0}\frac{dx}{\left ( 2x^{2}+1 \right )\sqrt{x^{2}+1}}
Substitute
x=\displaystyle \frac{1}{z}
\Rightarrow\displaystyle dx=-\frac{1}{z^2}dz
I=\displaystyle \int _{ \sqrt { 3 } }^{ \infty } \dfrac { -\dfrac { 1 }{ { z }^{ 2 } } dz }{ \left( \dfrac { 2 }{ { z }^{ 2 } } +1 \right) \sqrt { \dfrac { 1 }{ { z }^{ 2 } } +1 } }
=\displaystyle \int _{ \sqrt { 3 } }^{ \infty } \dfrac { -zdz }{ \left( 2+{ z }^{ 2 } \right) \sqrt { 1+{ z }^{ 2 } } }
Substitute
\sqrt { 1+{ z }^{ 2 } } =t
\Rightarrow z^2+1=t^2
\Rightarrow zdz=tdt
I=\displaystyle \int _{ 2 }^{ \infty } \frac { -tdt }{ \left( 1+{ t }^{ 2 } \right) t }
I=[\cot^{-1}t]_{2}^{\infty}
\Rightarrow I=-\cot^{-1}2
\Rightarrow I=-\tan^{-1}{\dfrac{1}{2}}
The value of definite integral
\displaystyle \int_{\infty }^{0}\frac{Ze^{-z}}{\sqrt{1-e^{-2z}}}dz
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\displaystyle -\frac{\pi }{2}ln2
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\displaystyle \frac{\pi }{2}ln2
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\displaystyle -\pi ln2
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\displaystyle \pi ln\frac{1}{\sqrt{2}}
Explanation
\displaystyle l=\int_{\infty }^{0}\frac{ze^{-z}}{\sqrt{1-e^{-2z}}}dz
put
\displaystyle e^{-z}=\sin \theta
\displaystyle l=-\int_{0}^{\pi /2}\frac{ln\left ( \sin \theta \right )\left ( -\cos \theta \right )d\theta }{\sqrt{1-\sin ^{2}\theta }}=\int_{0}^{\pi /2}ln\sin \theta d\theta
\displaystyle \frac{-\pi }{2}ln2
The derivative of
F(x)=\displaystyle\int^{x^3}_{x^2}\frac{dt}{log t}(x >0)
is
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\displaystyle\frac{1}{3log x}
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\displaystyle\frac{1}{3log x}-\frac{1}{2log x}
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(log x)^{-1}x(x-1)
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\displaystyle\frac{3x^2}{log x}
Explanation
Given,
F(x)=\displaystyle\int^{x^3}_{x^2}\frac{dt}{log t}
\Rightarrow F'(x)=\displaystyle\frac{d}{dx}F(x)
=\displaystyle\frac{d}{dx}(x^3)\cdot\frac{1}{log x^3}-\frac{d}{dx}(x^2)\cdot\frac{1}{log x^2}
=\displaystyle\frac{3x^2}{3log x}-\frac{2x}{2log x}=\frac{x^2-x}{log x}
=x(x-1)(log x)^{-1}
If
\displaystyle \int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } }\left( x-\alpha \right) } dx=0
, then
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1<\alpha <2
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\alpha <2
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0<\alpha<1
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\alpha=0
Explanation
\displaystyle\because \int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } }\left( x-\alpha \right) } dx=0,
\therefore { e }^{ { x }^{ 2 } }\left( x-\alpha \right)
must be
+ve
and
-ve
both for
x\in (0,1)
i.e.
{ e }^{ x }\left( x-\alpha \right) =0
for one
x\in (0,1)
\therefore \alpha \in(0,1)
Trapezoidal rule for evaluation of
\displaystyle\int _{ a }^{ b }{ f(x)dx }
requires the interval
(a,b)
to be divided into
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2n
sub-intervals of equal width
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any number of sub intervals of not equal width
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any number of sub intervals of equal width
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3n
sub-intervals of equal width
Explanation
Trapezoidal rule for evaluation of
\int _{ a }^{ b }{ f(x)dx }
requires the interval
(a,b)
to be divided into any number of sub-intervals of equal width.
\displaystyle \int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } }
is equal to
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\displaystyle \frac { \pi }{ 2 } +1
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\displaystyle \frac { \pi }{ 2 } -1
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\displaystyle 1-\frac { \pi }{ 2 }
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none of these
Explanation
Let
\displaystyle I=\int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } } =\int _{ 0 }^{ \pi /2 }{ \frac { a\cos { \theta } d\theta }{ a+a\cos { \theta } } }
By putting
x=a\sin { \theta } \Rightarrow dx=a\cos { \theta } d\theta
\displaystyle I=\int _{ 0 }^{ \pi /2 }{ \frac { \cos { \theta } }{ 1+\cos { \theta } } d\theta } =\int _{ 0 }^{ \pi /2 }{ \left( 1-\frac { 1 }{ 1+\cos { \theta } } \right) d\theta }
\displaystyle =\int _{ 0 }^{ \pi /2 }{ d\theta } -\frac { 1 }{ 2 } \int _{ 0 }^{ \pi /2 }{ \sec ^{ 2 }{ \frac { \theta }{ 2 } } d\theta }
\displaystyle ={ \left[ \theta -\tan { \frac { \theta }{ 2 } } \right] }_{ 0 }^{ \pi /2 }=\frac { \pi }{ 2 } -\tan { \frac { \pi }{ 4 } } =\left( \frac { \pi }{ 2 } -1 \right)
The value of the integral
\int_0^{\dfrac{\pi}{2}}\,sin^5x\,dx
is
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\dfrac{4}{15}
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\dfrac{8}{5}
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\dfrac{8}{15}
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\dfrac{4}{5}
Explanation
I=\int_0^{\dfrac{\pi}{2}}sin^4x.sin\,x\,dx
=\int_0^{\dfrac{\pi}{2}}(1-cos^2\,x)^2\,sin\,x\,dx
Put
cos\,x=t
\Rightarrow\;-sin\,x\,dx=dt
=-\int_1^0(1-t^2)^2dt=\int_0^(t^4-2t^2+1)dt
[\because\,-\int_a^bf(x)dx=\int_a^bf(x)dx]
=\dfrac{1}{5}(t^5)_0^1-\dfrac{2}{3}(t^3)_0^1+(t)_0^1
=\dfrac{1}{5}-\dfrac{2}{3}+1=\dfrac{3-10+15}{15}=\dfrac{8}{15}
Using Trapezoidal rule and following table
\int_0^8 {f(x)}dx
is equal to
x
0
2
4
6
8
f(x)
2
5
10
17
26
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184
0%
92
0%
46
0%
-36
\displaystyle \int_1^{\sqrt 3}\frac {dx}{1+x^2}
equals
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\dfrac {\pi}{3}
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\dfrac {2\pi}{3}
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\dfrac {\pi}{6}
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\dfrac {\pi}{12}
Explanation
\displaystyle \int \frac {d}{1+x^2}=\tan^{-1}x=F(x)
By second fundamental theorem of calculus, we obtain
\displaystyle \int_1^{\sqrt 3}\frac {dx}{1+x^2}=F(\sqrt 3)-F(1)
=\tan^{-1}\sqrt 3-\tan^{-1}1
\displaystyle =\frac {\pi}{3}-\frac {\pi}{4}=\frac {\pi}{12}
\displaystyle\int _{ 1 }^{ 3 }{ \dfrac { \cos { \left( \log { x } \right) } }{ x } dx }
is equal to
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1
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\cos { \left( \log { 3 } \right) }
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\sin { \left( \log { 3 } \right) }
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\dfrac { \pi }{ 4 }
Explanation
\displaystyle \int_{1}^{3} \frac{cos(logx)}{x}dx
Substituting
log\,x = t
\displaystyle dt = \frac{1}{x}dx
\displaystyle \therefore \int_{t_{1}}^{t_{2}} cos\,t\,dt
Lim
: t = log\,1 = 0
to
t = log\,3
\displaystyle \therefore \int_{0}^{log3} cos\,t\,dt \Rightarrow [sint]_{0}^{log3} = sin\,log3-0
\displaystyle \therefore \int_{1}^{3} \frac{cos(log\,x)}{x}dx = sin\,log3
The value of
I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( \tan ^{ n+1 }{ x } \right) dx } +\frac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( \tan ^{ n+1 }{ \left( \frac { x }{ 2 } \right) } \right) dx }
is
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\cfrac{1}{n}
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\cfrac{n+2}{2n+1}
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\cfrac{2n-1}{n}
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\cfrac{2n-3}{3n-2}
Explanation
Given
I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( \tan ^{ n+1 }{ x } \right) dx } +\cfrac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( \tan ^{ n+1 }{ \left( \cfrac { x }{ 2 } \right) } \right) dx }
In second integral, put
t=\cfrac{x}{2} \Rightarrow
dx=2dt
\Rightarrow
Also, when
x=0
then
t=0
When
x=\dfrac{\pi}2
then
t=\dfrac{\pi}4
Then
I=\displaystyle \int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n+1 }{ x } \right) } dx+\int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n-1 }{ t } \right) } dt
I=\displaystyle \int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n+1 }{ x } \right) } dx+\int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n-1 }{ x } \right) } dx\quad (\because \int _{ a }^{ b }{ f(x)dx } =\int _{ a }^{ b }{ f(y)dy } )
\Rightarrow I=\displaystyle \int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n+1 }{ x } +\tan ^{ n-1 }{ x } \right) } dx
\Rightarrow I=\displaystyle \int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n-1 }{ x } \right) .\left( \tan ^{ 2 }{ x+1 } \right) } dx
\Rightarrow I=\displaystyle \int _{ 0 }^{ \pi /4 }{ \left( \tan ^{ n-1 }{ x } \right) .\left( \sec ^{ 2 }{ x } \right) } dx
Put
t=\tan{x}
\Rightarrow dt=\sec ^{ 2 }{ x } dx
Also when
x=0
then
t=0
when
x=\pi /4
, then
t=1
I=\displaystyle \int _{ 0 }^{ 1 }{ { t }^{ n-1 }dt }
={ \left[ \cfrac { { t }^{ n } }{ n } \right] }_{ 0 }^{ 1 }=\cfrac { 1 }{ n }
What is
\displaystyle \int _{ 0 }^{ 1 }{ \cfrac { \tan ^{ -1 }{ x } }{ 1+{ x }^{ 2 } } } dx
equal to?
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\cfrac { { \pi }^{ 2 } }{ 8 }
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\cfrac { { \pi }^{ 2 } }{ 32 }
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\cfrac { \pi }{ 4 }
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\cfrac { \pi }{ 8 }
Explanation
I=\displaystyle \int _{ 0 }^{ 1 }{ \frac { \tan ^{ -1 }{ x } }{ 1+{ x }^{ 2 } } } dx
Put
\tan ^{ -1 }{ x } =p\Rightarrow \dfrac { dx }{ 1+{ x }^{ 2 } } =dp
When
x=0, p=0
and
x=1,p=\dfrac{\pi}{4}
\Rightarrow I=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 4 } }{ pdp }=\dfrac{p^{2}}{2} =\dfrac { { \pi }^{ 2 } }{ 32}
Hence, B is correct.
What is
\displaystyle\int _{ 1 }^{ 2 }{ \ln { x } dx }
equal to?
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\ln { 2 }
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1
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\ln { \left( \dfrac { 4 }{ e } \right) }
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\ln { \left( \dfrac { e }{ 4 } \right) }
Explanation
Let,
y=\ln x
then,
x={ e^{ y } }
Now at,
x=1,y=0
x=2,y=\ln 2
\Rightarrow dx={ e }^{ y }.dy
Putting the values, we get
\displaystyle \int_{1}^{2} \ln x dx=\int _{ 0 }^{ \ln2 }{ { e }^{ y }.y.dy }
=\displaystyle { \left[ y.{ e }^{ y } \right] }_{ 0 }^{\ln2 }-\int _{ 0 }^{\ln2 }{ { e }^{ y }dy }
=2\ln 2-1
=2\ln 2-\ln(e)
=\ln\left(\dfrac { 4 }{ e }\right)
Hence, C is correct.
If
\displaystyle \int_{0}^{b} \displaystyle \frac{dx}{1\, +\, x^2}\, =\, \displaystyle \int_{b}^{\infty} \displaystyle \frac{dx}{1\, +\, x^2}
, then
b =
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\tan ^{-1}\, \left (\displaystyle \frac{1}{3} \right )
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\displaystyle \frac{\sqrt 3}{2}
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\sqrt 3
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1
Explanation
\displaystyle \int_{0}^{b} \displaystyle \frac{dx}{1\, +\, x^2}\, =\, \displaystyle \int_{b}^{\infty} \displaystyle \frac{dx}{1\, +\, x^2}
\Rightarrow [\tan^{-1}x]_0^b=[\tan^{-1}x]_b^{\infty}
\Rightarrow \tan^{-1}b-0=\dfrac{\pi}{2}-\tan^{-1}b
\Rightarrow \tan^{-1}b=\dfrac{\pi}{4}
\Rightarrow b=\tan\dfrac{\pi}{4}=1
Select and write the most appropriate answer from the given alternatives for question :
If
\displaystyle \int^k_0 4x^3dx=16
, then the value of
k
is _____.
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1
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2
0%
3
0%
4
Explanation
Given:
\displaystyle \int^K_0 4x^3dx=16
4\left(\dfrac{x^4}{4}\right)^K_0=16
(x^4)^k_0=16
k^4=16
k^4=2^4
\boxed{k=2}
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Practice Class 12 Commerce Applied Mathematics Quiz Questions and Answers
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