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Question
Mathematics
If ∫ limits0π (5 cos x(1+ cos x cos 3 x+ cos 2 x+ cos 3 x cos 3 x) d x/1+5 cos x)=( k π/16), then k is equal to
Q. If
0
∫
π
1
+
5
c
o
s
x
5
c
o
s
x
(
1
+
c
o
s
x
c
o
s
3
x
+
c
o
s
2
x
+
c
o
s
3
x
c
o
s
3
x
)
d
x
=
16
kπ
, then
k
is equal to
375
156
JEE Main
JEE Main 2023
Integrals
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Answer:
26
Solution:
I
=
0
∫
π
1
+
5
c
o
s
x
5
c
o
s
x
(
1
+
c
o
s
x
c
o
s
3
x
+
c
o
s
2
x
+
c
o
s
3
x
c
o
s
3
x
)
d
x
I
=
0
∫
π
1
+
5
−
c
o
s
x
5
−
c
o
s
x
(
1
+
c
o
s
x
c
o
s
3
x
+
c
o
s
2
x
+
c
o
s
3
x
c
o
s
3
x
)
d
x
2
I
=
0
∫
π
(
1
+
cos
x
cos
3
x
+
cos
2
x
+
cos
3
x
cos
3
x
)
d
x
2
I
=
2
0
∫
2
π
(
1
+
cos
x
cos
3
x
+
cos
2
x
+
cos
3
x
cos
3
x
)
d
x
I
=
0
∫
2
π
(
1
+
sin
x
(
−
sin
3
x
)
+
sin
2
x
−
sin
3
x
sin
3
x
)
d
x
2
I
=
0
∫
2
π
(
3
+
cos
4
x
+
cos
3
x
cos
3
x
−
sin
3
x
sin
3
x
)
d
x
2
I
=
0
∫
2
π
3
+
cos
4
x
+
(
4
c
o
s
3
x
+
3
c
o
s
x
)
cos
3
x
−
sin
3
x
(
4
3
s
i
n
x
−
s
i
n
3
x
)
d
x
2
I
=
0
∫
2
π
(
3
+
cos
4
x
+
4
1
+
4
3
cos
4
x
)
d
x
2
I
=
4
13
×
2
π
+
4
7
(
4
s
i
n
4
x
)
0
2
π
⇒
I
=
16
13
π