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Byju's Answer
Standard XII
Mathematics
Property 1
Show that ∫...
Question
Show that
∫
d
x
sin
x
(
5
+
4
cos
x
)
=
1
18
log
(
1
−
cos
x
)
−
1
2
log
(
1
+
cos
x
)
+
4
9
log
(
5
+
4
cos
x
)
.
Open in App
Solution
Let
I
=
∫
1
sin
x
(
5
+
4
cos
x
)
d
x
Multiplying numerator and denominator by
sin
x
, we get
I
=
−
∫
sin
x
(
4
cos
x
+
5
)
(
cos
2
x
−
1
)
d
x
Put
t
=
cos
x
⇒
d
t
=
−
sin
x
d
x
So,
I
=
∫
1
(
4
t
+
5
)
(
t
2
−
1
)
d
t
=
1
(
4
t
+
5
)
(
t
+
1
)
(
t
−
1
)
Resolving
1
(
4
t
+
5
)
(
t
+
1
)
(
t
−
1
)
into partial fractions
1
(
4
t
+
5
)
(
t
+
1
)
(
t
−
1
)
=
A
4
t
+
5
+
B
t
+
1
+
C
t
−
1
....(1)
⇒
1
=
A
(
t
2
−
1
)
+
B
(
t
−
1
)
(
4
t
+
5
)
+
C
(
4
t
+
5
)
(
t
+
1
)
....(2)
Put
t
=
1
in eqn (2)
⇒
C
=
1
18
Put
t
=
−
1
ineqn (2)
⇒
B
=
−
1
2
Put
t
=
−
5
4
⇒
A
=
16
9
Put these values in eqn (1)
1
(
4
t
+
5
)
(
t
+
1
)
(
t
−
1
)
=
16
9
(
4
t
+
5
)
+
−
1
2
(
t
+
1
)
+
1
18
(
t
−
1
)
I
=
∫
1
(
4
t
+
5
)
(
t
2
−
1
)
d
t
=
16
9
∫
d
t
(
4
t
+
5
)
−
1
2
∫
d
t
(
t
+
1
)
+
1
18
∫
d
t
(
t
−
1
)
=
−
1
2
log
(
t
+
1
)
+
16
9
×
1
4
log
(
4
t
+
5
)
+
1
18
log
(
t
−
1
)
+
C
=
−
1
2
log
(
cos
x
+
1
)
+
4
9
log
(
4
cos
x
+
5
)
+
1
18
log
(
cos
x
−
1
)
+
C
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0
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