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CBSE Questions for Class 12 Commerce Maths Integrals Quiz 4 - MCQExams.com
CBSE
Class 12 Commerce Maths
Integrals
Quiz 4
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
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
.
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
)
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
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
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
)
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
∫
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
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
Solve
∫
−
13
2
d
x
5
√
(
3
−
x
)
4
Report Question
0%
−
5
(
5
√
16
−
1
)
0%
5
(
5
√
16
−
1
)
0%
−
5
(
5
√
16
+
1
)
0%
None of these
Explanation
Let
I
=
∫
−
13
2
d
x
5
√
(
3
−
x
)
4
Put
3
−
x
=
t
⇒
−
d
x
=
d
t
I
=
∫
16
1
d
t
t
4
5
=
[
t
1
5
1
5
]
16
1
=
−
5
(
5
√
16
−
1
)
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
b
→
∞
1
2
log
(
1
+
x
2
)
−
1
2
log
(
1
+
x
)
+
1
2
t
a
n
−
1
x
)
b
0
=
lim
b
→
∞
(
1
2
log
(
1
+
b
)
−
1
2
log
(
1
+
b
)
+
1
2
t
a
n
−
1
b
)
=
π
4
State true or false:
The average value of the function
f
(
x
)
=
s
i
n
2
x
c
o
s
3
x
on the interval
[
−
π
,
π
]
is 0.
Report Question
0%
True
0%
False
Explanation
∫
π
−
π
f
(
x
)
d
x
=
∫
π
−
π
sin
2
x
cos
3
x
d
x
=
∫
π
−
π
sin
2
x
(
1
−
sin
2
x
)
cos
x
d
x
=
∫
π
−
π
sin
2
x
cos
x
d
x
−
∫
π
−
π
sin
4
x
cos
x
d
x
=
[
sin
3
x
3
]
π
−
π
−
[
sin
5
x
3
]
π
−
π
=
0
I
1
is equal to
Report Question
0%
2
3
∫
π
/
2
0
(
s
i
n
2
θ
)
(
c
o
s
θ
)
−
1
/
3
d
θ
0%
3
2
∫
π
/
2
0
(
s
i
n
2
θ
)
(
c
o
s
θ
)
−
1
/
3
d
θ
0%
2
3
∫
π
/
2
0
(
s
i
n
θ
)
2
/
3
(
c
o
s
θ
)
−
1
/
3
d
θ
0%
3
2
∫
π
/
2
0
(
s
i
n
θ
)
2
/
3
(
c
o
s
θ
)
−
1
/
3
d
θ
Explanation
I
1
=
∫
1
0
(
1
−
x
2
)
1
/
3
d
x
=
(
1
−
x
2
)
3
0
+
2
3
∫
1
0
x
2
(
1
−
x
2
)
2
/
3
d
x
x
=
s
i
n
θ
,
d
x
=
c
o
s
θ
d
θ
I
1
=
2
3
∫
π
/
2
0
s
i
n
2
θ
c
o
s
θ
(
c
o
s
θ
)
4
/
3
d
θ
I
1
=
2
3
∫
π
/
2
0
s
i
n
2
θ
(
c
o
s
θ
)
−
1
/
3
d
θ
Ans:
A
Evaluate
∫
π
/
2
0
d
x
2
+
sin
2
x
Report Question
0%
2
π
3
0%
π
3
0%
2
π
5
0%
None of these
Explanation
∫
2
π
0
d
x
2
+
sin
2
x
=
∫
2
π
0
d
x
2
+
2
tan
x
1
+
tan
2
x
d
x
=
1
2
∫
2
x
0
sec
2
x
tan
2
x
+
tan
x
+
1
d
x
Put
tan
x
=
t
⇒
sec
2
x
d
x
=
d
t
I
=
1
2
∫
1
t
2
+
t
+
1
d
t
.
=
1
2
∫
1
(
t
+
1
2
)
2
−
3
2
=
2
π
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
x
=
1
t
⇒
d
x
=
−
1
t
2
d
t
l
=
−
∫
1
/
3
3
t
c
o
s
e
c
99
(
1
t
−
t
)
1
t
2
d
t
l
=
−
∫
3
1
/
3
1
t
c
o
s
e
c
99
(
t
−
1
t
)
d
t
l
=
−
1
⇒
2
l
=
0
⇒
l
=
0
∫
0
1
√
3
d
x
(
2
x
2
+
1
)
√
x
2
+
1
Report Question
0%
−
tan
−
1
1
2
0%
tan
−
1
1
0%
−
tan
−
1
1
3
0%
tan
−
1
1
√
2
Explanation
I
=
∫
0
1
√
3
d
x
(
2
x
2
+
1
)
√
x
2
+
1
Substitute
x
=
1
z
⇒
d
x
=
−
1
z
2
d
z
I
=
∫
∞
√
3
−
1
z
2
d
z
(
2
z
2
+
1
)
√
1
z
2
+
1
=
∫
∞
√
3
−
z
d
z
(
2
+
z
2
)
√
1
+
z
2
Substitute
√
1
+
z
2
=
t
⇒
z
2
+
1
=
t
2
⇒
z
d
z
=
t
d
t
I
=
∫
∞
2
−
t
d
t
(
1
+
t
2
)
t
I
=
[
cot
−
1
t
]
∞
2
⇒
I
=
−
cot
−
1
2
⇒
I
=
−
tan
−
1
1
2
The value of definite integral
∫
0
∞
Z
e
−
z
√
1
−
e
−
2
z
d
z
Report Question
0%
−
π
2
l
n
2
0%
π
2
l
n
2
0%
−
π
l
n
2
0%
π
l
n
1
√
2
Explanation
l
=
∫
0
∞
z
e
−
z
√
1
−
e
−
2
z
d
z
put
e
−
z
=
sin
θ
l
=
−
∫
π
/
2
0
l
n
(
sin
θ
)
(
−
cos
θ
)
d
θ
√
1
−
sin
2
θ
=
∫
π
/
2
0
l
n
sin
θ
d
θ
−
π
2
l
n
2
If
∫
1
0
e
x
2
(
x
−
α
)
d
x
=
0
, then
Report Question
0%
1
<
α
<
2
0%
α
<
2
0%
0
<
α
<
1
0%
α
=
0
Explanation
∵
∫
1
0
e
x
2
(
x
−
α
)
d
x
=
0
,
∴
e
x
2
(
x
−
α
)
must be
+
v
e
and
−
v
e
both for
x
∈
(
0
,
1
)
i.e.
e
x
(
x
−
α
)
=
0
for one
x
∈
(
0
,
1
)
∴
α
∈
(
0
,
1
)
∫
a
0
d
x
a
+
√
a
2
−
x
2
is equal to
Report Question
0%
π
2
+
1
0%
π
2
−
1
0%
1
−
π
2
0%
none of these
Explanation
Let
I
=
∫
a
0
d
x
a
+
√
a
2
−
x
2
=
∫
π
/
2
0
a
cos
θ
d
θ
a
+
a
cos
θ
By putting
x
=
a
sin
θ
⇒
d
x
=
a
cos
θ
d
θ
I
=
∫
π
/
2
0
cos
θ
1
+
cos
θ
d
θ
=
∫
π
/
2
0
(
1
−
1
1
+
cos
θ
)
d
θ
=
∫
π
/
2
0
d
θ
−
1
2
∫
π
/
2
0
sec
2
θ
2
d
θ
=
[
θ
−
tan
θ
2
]
π
/
2
0
=
π
2
−
tan
π
4
=
(
π
2
−
1
)
What is
∫
2
1
ln
x
d
x
equal to?
Report Question
0%
ln
2
0%
1
0%
ln
(
4
e
)
0%
ln
(
e
4
)
Explanation
Let,
y
=
ln
x
then,
x
=
e
y
Now at,
x
=
1
,
y
=
0
x
=
2
,
y
=
ln
2
⇒
d
x
=
e
y
.
d
y
Putting the values, we get
∫
2
1
ln
x
d
x
=
∫
ln
2
0
e
y
.
y
.
d
y
=
[
y
.
e
y
]
ln
2
0
−
∫
ln
2
0
e
y
d
y
=
2
ln
2
−
1
=
2
ln
2
−
ln
(
e
)
=
ln
(
4
e
)
Hence, C is correct.
∫
π
0
1
1
+
sin
x
d
x
is equal to
Report Question
0%
1
0%
2
0%
3
0%
4
Explanation
Let
I
=
∫
π
0
1
1
+
sin
x
d
x
=
∫
π
0
1
1
+
2
tan
x
2
1
+
tan
2
x
2
d
x
=
∫
π
0
sec
2
x
2
(
1
+
tan
x
2
)
2
d
x
Put
1
+
tan
x
2
=
t
⇒
1
2
sec
2
x
2
=
d
t
∴
I
=
∫
∞
1
2
d
t
1
+
t
2
=
−
2
t
]
∞
1
=
2
The value of
∫
x
/
4
0
sec
x
(
sec
x
+
tan
x
)
2
d
x
is
Report Question
0%
1
+
√
2
0%
−
11
+
√
2
0%
−
√
2
0%
None of these
Explanation
I
=
∫
π
4
0
sec
x
(
sec
x
+
tan
x
)
2
d
x
=
∫
π
4
0
1
/
cos
x
[
1
cos
x
+
sin
x
cos
x
]
2
d
x
=
∫
π
4
0
cos
2
x
/
cos
x
(
1
+
sin
x
)
2
d
x
I
=
∫
π
4
0
(
1
+
sin
x
)
−
2
cos
x
d
x
Take
1
+
sin
x
=
t
⟹
cos
x
d
x
=
d
t
When
x
=
0
,
t
=
1
and
x
=
π
4
,
t
=
1
+
1
√
2
∴
I
=
∫
1
+
1
√
2
1
t
−
2
d
t
∴
I
=
|
t
−
1
−
1
|
1
+
1
√
2
1
=
−
[
1
1
+
1
√
2
−
1
]
=
−
√
2
√
2
+
1
+
1
=
−
√
2
+
√
2
+
1
√
2
+
1
=
1
√
2
+
1
=
1
√
2
+
1
∗
√
2
−
1
√
2
−
1
=
√
2
−
1
2
−
1
=
√
2
−
1
∴
None of these is correct option
If
f
(
x
)
=
{
2
x
2
+
1
,
x
≤
1
4
x
2
−
1
,
x
>
1
, then
∫
2
0
f
(
x
)
d
x
is
Report Question
0%
10
0%
50
/
3
0%
1
/
3
0%
47
/
2
Explanation
f
(
x
)
=
{
2
x
2
+
1
,
x
≤
1
4
x
2
−
1
,
x
>
1
from 0 to 1 we have
2
x
2
+
1
and from 1 to 2 we have
4
x
2
−
1
=
∫
1
0
(
2
x
2
+
1
)
d
x
+
∫
2
1
(
4
x
2
−
1
)
d
x
=
[
2
x
3
3
+
x
]
1
0
+
[
4
x
3
3
−
x
]
2
1
=
2
3
+
1
−
0
−
0
+
4
(
2
)
3
3
−
2
−
(
4
3
−
1
)
=
2
3
+
1
+
32
3
−
2
−
4
3
+
1
=
2
+
32
−
4
3
=
30
3
=10.
What is
∫
1
0
tan
−
1
x
1
+
x
2
d
x
equal to ?
Report Question
0%
π
4
0%
π
8
0%
π
2
8
0%
π
2
32
Explanation
Let
I
=
∫
1
0
tan
−
1
x
1
+
x
2
d
x
Let,
y
=
tan
−
1
x
⇒
d
y
=
1
1
+
x
2
d
x
Now for
x
=
1
,
y
=
π
4
and for
x
=
0
,
y
=
0
Putting the values in the integral we have,
I
=
∫
π
4
0
y
.
d
y
=
[
y
2
2
]
π
4
0
=
π
2
32
Hence, D is correct.
If
d
y
d
t
=
k
y
and
k
≠
0
, which of the following could be the equation of
y
?
Report Question
0%
y
=
k
x
−
7
0%
y
=
95
e
k
t
0%
y
=
5
+
l
n
k
0%
y
=
(
x
−
k
)
2
0%
y
=
k
√
x
Explanation
Given,
d
y
d
t
=
k
y
and
k
≠
0
We have
d
y
y
=
k
d
t
Apply integral on both sides, so we get
∫
d
y
y
=
∫
k
d
t
⇒
ln
(
y
)
=
k
t
+
c
(where
c
is constant)
⇒
y
=
e
c
e
k
t
=
a
e
k
t
(
a
=
e
c
)
The possible solution is
y
=
95
e
k
t
Solve
∫
π
2
0
√
sin
ϕ
cos
5
ϕ
d
ϕ
.
Report Question
0%
64
231
0%
24
231
0%
54
231
0%
None of these
Explanation
Let
I
=
∫
π
2
0
√
sin
ϕ
cos
5
ϕ
d
ϕ
=
∫
π
2
0
√
sin
ϕ
cos
4
ϕ
cos
ϕ
d
ϕ
=
∫
π
2
0
√
sin
ϕ
(
1
−
sin
2
ϕ
)
2
cos
ϕ
d
ϕ
p
u
t
t
=
sin
ϕ
d
t
d
ϕ
=
cos
ϕ
d
t
=
cos
ϕ
d
ϕ
W
h
e
n
ϕ
=
0
,
t
=
0
w
h
e
n
ϕ
=
π
2
,
t
=
1
∴
I
=
∫
1
0
√
t
(
1
−
t
2
)
2
d
t
=
∫
1
0
√
t
(
1
−
2
t
2
+
t
4
)
d
t
=
∫
1
0
t
1
2
+
t
9
2
−
2
t
5
2
d
t
=
[
2
3
t
3
2
+
2
11
t
11
2
−
4
7
t
7
2
]
1
0
=
2
3
+
2
11
−
4
7
=
154
+
42
−
132
3
×
11
×
7
=
64
231
∫
√
π
/
2
0
2
x
3
sin
(
x
2
)
d
x
is equal to
Report Question
0%
1
√
2
(
1
+
π
4
)
0%
1
√
2
(
1
−
π
4
)
0%
1
√
2
(
π
2
−
1
)
0%
1
√
2
(
1
−
π
2
)
0%
1
√
2
(
π
4
−
1
)
Explanation
I
=
∫
√
π
/
2
0
2
x
3
sin
(
x
2
)
d
x
=
∫
√
π
/
2
0
2
x
2
⋅
x
sin
(
x
2
)
d
x
Put
x
2
=
t
⇒
2
x
d
x
=
d
t
Also, when
x
=
0
, then
t
=
0
and when
x
=
√
π
2
, then
t
=
π
4
⇒
I
=
∫
π
/
4
0
t
I
sin
t
I
I
d
t
=
[
t
(
−
cos
t
)
]
π
/
4
0
−
∫
π
/
4
0
−
cos
t
(
1
)
d
t
=
[
−
t
cos
t
]
π
/
4
0
+
∫
π
/
4
0
cos
t
d
t
=
[
−
t
cos
t
+
sin
t
]
π
/
4
0
=
[
−
π
4
⋅
1
√
2
+
1
√
2
]
=
1
√
2
(
1
−
π
4
)
∫
1
7
sin
(
x
7
+
10
)
d
x
is equal to
Report Question
0%
1
7
cos
(
x
7
+
10
)
+
C
0%
−
1
7
cos
(
x
7
+
10
)
+
C
0%
−
cos
(
x
7
+
10
)
+
C
0%
−
7
cos
(
x
7
+
10
)
+
C
0%
cos
(
x
+
70
)
+
C
Explanation
Let
I
=
∫
1
7
sin
(
x
7
+
10
)
d
x
=
1
7
∫
sin
(
x
7
+
10
)
=
1
7
−
cos
(
x
7
+
10
)
1
7
=
−
cos
(
x
7
+
10
)
+
C
If
∫
1
0
x
d
x
=
π
4
−
1
2
l
n
2
then the value of definite integral
∫
1
0
tan
−
1
(
1
−
x
+
x
2
)
d
x
equals :
Report Question
0%
l
n
2
0%
π
4
+
l
n
2
0%
π
4
−
l
n
2
0%
2
l
n
2
The value of
∫
1
0
8
log
(
1
+
x
)
1
+
x
2
d
x
is
Report Question
0%
π
2
log
2
0%
π
log
2
0%
2
π
log
2
0%
None of these
Explanation
Let
I
=
∫
1
0
8
log
(
1
+
x
)
1
+
x
2
d
x
=
8
∫
1
0
log
(
1
+
x
)
1
+
x
2
d
x
p
u
t
x
=
tan
θ
⇒
d
x
=
sin
2
θ
d
θ
x
,
π
4
,
x
=
0
I
=
8
∫
π
4
0
log
(
1
+
x
)
sin
2
θ
sin
2
θ
d
θ
=
8
∫
π
4
0
log
(
1
+
tan
x
)
d
x
=
8
×
π
8
log
2
=
π
log
2
Hence option (A) is true.
If
∫
√
1
+
sin
x
⋅
f
(
x
)
d
x
=
2
3
(
1
+
sin
x
)
3
/
2
+
C
, then
f
(
x
)
is equal to
Report Question
0%
cos
x
0%
sin
x
0%
tan
x
0%
1
Explanation
∫
√
1
+
sin
x
.
f
(
x
)
d
x
=
2
3
(
1
+
sin
x
)
3
/
2
+
c
On differentiating both sides, we get
√
1
+
sin
x
.
f
(
x
)
=
2
3
.
3
2
(
1
+
sin
x
)
1
/
2
.
cos
x
+
0
⇒
f
(
x
)
=
cos
x
∫
π
/
2
0
f
(
sin
2
x
)
sin
x
d
x
=
K
∫
π
/
2
0
f
(
cos
2
x
)
cos
x
d
x
where
k
equals to
Report Question
0%
2
0%
4
0%
√
2
0%
2
√
2
Explanation
∫
π
2
0
sin
2
x
sin
x
d
x
=
K
∫
π
2
0
cos
2
x
cos
x
d
x
∫
π
2
0
2
sin
x
cos
x
sin
x
d
x
=
K
∫
π
2
0
(
1
−
2
sin
2
x
)
cos
x
d
x
∫
π
2
0
2
sin
2
x
cos
x
d
x
=
K
[
∫
π
2
0
cos
x
d
x
−
∫
π
2
0
2
sin
2
x
cos
x
d
x
]
{
2
sin
3
x
3
}
π
2
0
=
K
[
{
sin
x
}
π
2
0
−
{
2
sin
3
x
3
}
π
2
0
]
2
3
=
K
[
1
−
2
3
]
K
=
2
∫
1
0
d
x
x
√
x
Report Question
0%
2
0%
−
2
0%
1
0%
3
Explanation
∫
1
0
1
x
√
x
d
x
∫
1
0
x
−
3
2
d
x
=
[
x
−
3
2
+
1
−
3
2
+
1
]
1
0
=
[
x
−
1
2
−
1
2
]
1
0
=
1
−
0
−
1
2
=
−
2
What is
∫
2
π
0
√
1
+
sin
x
2
d
x
equal to?
Report Question
0%
8
0%
4
0%
2
0%
0
Explanation
1
+
sin
x
2
=
sin
2
x
4
+
cos
2
x
4
+
2
sin
x
4
cos
x
4
(
sin
x
4
+
cos
x
4
)
2
√
1
+
sin
x
2
=
√
(
sin
x
4
+
cos
x
4
)
2
=
sin
x
4
+
cos
x
4
∫
2
π
0
(
sin
x
4
+
cos
x
4
)
d
x
=
4
(
−
cos
x
4
+
sin
x
4
)
+
c
A
f
t
e
r
p
u
t
t
i
n
g
t
h
e
l
i
m
i
t
w
e
g
e
t
⇒
4
(
1
−
(
−
1
)
)
=
8
So correct answer will be option A
Let
I
=
∫
π
/
3
π
/
4
sin
x
x
d
x
. Then?
Report Question
0%
1
2
≤
I
≤
1
0%
4
≤
I
≤
2
√
30
0%
√
3
8
≤
I
≤
√
2
6
0%
1
≤
I
≤
2
√
3
√
2
Explanation
I
=
∫
π
3
π
4
s
i
n
x
x
d
x
s
i
n
x
x
is a decreasing function in given interval
difference of limits
=
π
3
−
π
4
=
π
12
so,
π
12
⋅
s
i
n
π
3
π
3
≤
I
≤
π
12
⋅
s
i
n
π
4
π
4
√
3
8
≤
I
≤
√
2
6
∫
(
e
x
)
x
(
2
+
log
x
)
d
x
=
.
.
.
.
+
c
,
x
∈
R
+
−
{
1
}
Report Question
0%
x
x
0%
(
e
x
)
x
0%
e
x
0%
(
1
+
log
x
)
(
e
x
)
x
Explanation
∫
(
e
x
)
x
(
2
+
log
x
)
d
x
=
∫
(
e
x
)
x
d
x
+
∫
e
x
.
x
x
(
1
+
log
x
)
d
x
Integrating by parts ,
⇒
∫
(
e
x
)
x
+
e
x
(
∫
x
x
(
1
+
log
x
)
)
−
∫
(
d
d
x
(
e
x
)
.
(
∫
x
x
(
1
+
log
x
)
d
x
)
)
d
x
⇒
∫
(
e
x
)
x
+
e
x
x
x
−
∫
(
e
x
)
x
⇒
(
e
x
)
x
If
I
=
1
∫
0
x
(
1
−
x
)
1
/
2
d
x
and
60
I
+
k
=
25
then
k
=
_________.
(
k
∈
R
)
.
Report Question
0%
9
0%
25
0%
60
0%
41
Explanation
We know,
∫
a
0
f
(
x
)
d
x
=
∫
a
0
f
(
a
−
x
)
d
x
∴
I
=
∫
1
0
x
(
1
−
x
)
1
/
2
d
x
=
∫
1
0
(
1
−
x
)
x
1
/
2
d
x
⇒
I
=
∫
1
0
(
x
1
/
2
−
x
3
/
2
)
d
x
⇒
I
=
[
2
3
x
3
/
2
−
2
5
x
5
/
2
]
1
0
⇒
I
=
2
3
−
2
5
=
4
15
60
I
+
k
=
25
k
=
25
−
60
×
4
15
=
9
∫
32
1
d
x
x
1
/
5
√
1
+
x
4
/
5
Report Question
0%
2
5
(
√
17
+
√
2
)
0%
2
5
(
√
17
−
√
2
)
0%
5
2
(
√
17
−
√
2
)
0%
5
2
(
√
17
+
√
2
)
∫
π
/
2
0
sin
2
x
.
sin
x
d
x
=
.
.
.
.
.
Report Question
0%
1
3
0%
2
3
0%
−
2
3
0%
4
3
Explanation
I
=
∫
π
/
2
0
sin
2
x
.
sin
x
d
x
=
∫
π
/
2
0
2
sin
x
cos
x
.
sin
x
d
x
=
2
∫
π
/
2
0
sin
2
x
d
d
x
sin
x
d
x
=
2
3
[
sin
2
x
]
π
/
2
0
=
2
3
(
1
−
0
)
=
2
3
Evaluate
∫
5
π
−
2
π
cot
−
1
(
tan
x
)
d
x
.
Report Question
0%
0
0%
−
1
0%
1
0%
2
Explanation
I
=
∫
5
π
2
π
cot
−
1
(
tan
x
)
d
x
......
(
1
)
Applying property of limits of integral formula ;
⇒
I
=
∫
5
π
2
π
cot
−
1
(
tan
(
7
π
−
x
)
)
d
x
⇒
I
=
∫
5
π
2
π
cot
−
1
(
−
tan
x
)
d
x
⇒
I
=
−
∫
5
π
2
π
cot
−
1
(
tan
x
)
d
x
......
(
2
)
Adding both equations;
⇒
2
I
=
0
⇒
I
=
0
⇒
∫
5
π
2
π
cot
−
1
(
tan
x
)
d
x
=
0
∫
(
3.
x
2
.
tan
−
1
x
+
x
3
1
+
x
2
)
d
x
=
.
.
.
.
+
c
.
Report Question
0%
x
3
tan
−
1
x
0%
x
3
3
tan
−
1
x
0%
x
2
tan
−
1
x
0%
x
2
2
tan
−
1
x
Explanation
Given
∫
(
3
x
2
.
tan
−
1
x
+
x
3
1
+
x
2
)
d
x
=
∫
(
3
x
2
.
tan
−
1
x
.
d
x
)
+
∫
(
x
3
1
+
x
2
)
d
x
Integrating by parts
∫
3
x
2
t
a
n
−
1
x
.
d
x
+
x
3
∫
1
1
+
x
2
d
x
−
∫
(
d
d
x
(
x
3
)
(
∫
1
1
+
x
2
d
x
)
d
x
)
d
d
x
x
3
=
3
x
2
;
∫
1
1
+
x
2
=
t
a
n
−
1
x
⇒
∫
3
x
2
t
a
n
−
1
x
.
d
x
+
x
3
tan
−
1
x
−
∫
3
x
2
t
a
n
−
1
x
.
d
x
⇒
x
3
tan
−
1
x
+
c
If
∫
π
/
2
0
sin
x
cos
x
d
x
is equal to:
Report Question
0%
1
2
0%
1
4
0%
2
0%
1
Explanation
∫
π
/
2
0
sin
x
cos
x
d
x
sin
x
=
t
⟹
cos
x
d
x
=
d
t
x
→
0
→
π
2
t
→
0
→
1
⇒
∫
π
/
2
0
t
d
t
⇒
t
2
2
|
1
0
⇒
1
2
−
0
=
1
2
I
=
∫
x
+
2
(
x
+
1
)
2
d
x
;
then I is equal to
Report Question
0%
log
(
x
+
1
)
+
1
x
+
1
+
c
0%
log
(
x
+
2
)
−
1
x
+
1
+
c
0%
log
(
1
+
x
)
−
1
x
+
1
+
c
0%
log
(
x
+
2
)
+
1
x
+
1
+
c
Explanation
∫
x
+
2
(
x
+
1
)
2
d
x
=
∫
x
+
1
+
1
(
x
+
1
)
2
d
x
=
∫
(
x
+
1
(
x
+
1
)
2
+
1
(
x
+
1
)
2
)
d
x
=
∫
1
x
+
1
d
x
+
∫
1
(
x
+
1
)
2
d
x
[
∵
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
c
]
=
log
(
x
+
1
)
+
(
x
+
1
)
−
2
+
1
−
2
+
1
+
c
[
∵
∫
1
x
d
x
=
log
x
+
c
]
=
log
(
x
+
1
)
+
(
−
1
)
1
(
x
+
1
)
+
c
=
log
(
x
+
1
)
−
1
x
+
1
+
c
Hence, the answer is
log
(
x
+
1
)
−
1
x
+
1
+
c
.
If
∫
π
/
3
0
cos
x
3
+
4
sin
x
d
x
=
k
log
(
3
+
2
√
3
3
)
, then,
k
is equal to ?
Report Question
0%
1
2
0%
1
3
0%
1
4
0%
1
8
Explanation
sin
x
=
t
⇒
d
x
.
cos
x
=
d
t
I
=
∫
π
3
0
d
t
3
+
4
t
=
1
4
[
l
n
(
3
+
4
t
)
]
π
3
When
x
=
0
⇒
t
=
0
When
x
=
π
3
⇒
t
=
√
3
2
=
1
4
[
l
n
(
3
+
4
t
)
]
√
3
2
−
1
4
[
l
n
(
3
+
4
t
)
]
0
=
1
4
l
n
(
3
+
2
√
3
)
−
1
4
l
n
(
3
)
I
=
1
4
l
n
[
3
+
2
√
3
3
]
K
=
1
4
Find proper substitution
∫
1
0
e
−
x
1
+
e
−
x
d
x
Report Question
0%
1
+
e
−
x
→
t
0%
−
e
−
x
d
x
→
d
t
0%
−
∫
1
0
d
t
t
0%
−
∫
1
0
l
n
|
t
|
Explanation
∫
1
0
e
−
x
1
+
e
−
x
d
x
1
+
e
−
x
=
t
⇒
−
e
−
x
d
x
=
d
t
∫
1
0
−
d
t
t
∴
1
+
e
−
x
=
t
0:0:1
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0
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1
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Incorrect : 0
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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