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How to integrate dx/(sin(x)+sin(2x)) ?
Calculus
1 Answer
Noah G · maganbhai P.
Feb 23, 2018
I
=
1
2
ln
|
cos
x
+
1
|
+
1
6
ln
|
cos
x
−
1
|
−
2
3
ln
|
2
cos
x
+
1
|
+
C
Explanation:
Using some trig:
I
=
∫
1
sin
x
+
2
sin
x
cos
x
d
x
I
=
∫
1
sin
x
(
1
+
2
cos
x
)
d
x
I
=
∫
sin
x
sin
2
x
(
1
+
2
cos
x
)
d
x
I
=
∫
−
sin
x
(
cos
2
x
−
1
)
(
1
+
2
cos
x
)
d
x
I
=
∫
−
sin
x
(
cos
x
+
1
)
(
cos
x
−
1
)
(
2
cos
x
+
1
)
d
x
Now we let
u
=
cos
x
. Then
d
u
=
−
sin
x
d
x
and
d
x
=
d
u
−
sin
x
.
I
=
∫
−
sin
x
(
cos
x
+
1
)
(
cos
x
−
1
)
(
2
cos
x
+
1
)
⋅
d
u
−
sin
x
I
=
∫
1
(
u
+
1
)
(
u
−
1
)
(
2
u
+
1
)
d
u
This becomes a partial fraction problem.
A
u
+
1
+
B
u
−
1
+
C
2
u
+
1
=
1
(
u
+
1
)
(
u
−
1
)
(
2
u
+
1
)
A
(
u
−
1
)
(
2
u
+
1
)
+
B
(
u
+
1
)
(
2
u
+
1
)
+
C
(
u
+
1
)
(
u
−
1
)
=
1
A
(
2
u
2
−
2
u
+
u
−
1
)
+
B
(
2
u
2
+
2
u
+
u
+
1
)
+
C
(
u
2
−
1
)
=
1
A
(
2
u
2
−
u
−
1
)
+
B
(
2
u
2
+
3
u
+
1
)
+
C
(
u
2
−
1
)
=
1
2
A
u
2
−
A
u
−
A
+
2
B
u
2
+
3
B
u
+
B
+
C
u
2
−
C
=
1
Thus
⎧
⎪
⎨
⎪
⎩
2
A
+
2
B
+
C
=
0
3
B
−
A
=
0
B
−
A
−
C
=
1
If we substitute the second equation
A
=
3
B
into the first and third we get.
3...
B
−
(
3
B
)
−
C
=
1
−
2
B
−
C
=
1
1...
2
(
3
B
)
+
2
B
+
C
=
0
6
B
+
2
B
+
C
=
0
8
B
+
C
=
0
We can easily solve this system by elimination .
6
B
=
1
B
=
1
6
Therefore
8
(
1
6
)
+
C
=
0
C
=
−
4
3
And
A
=
3
B
=
3
(
1
6
)
=
1
2
Thus the partial fraction decomposition is as follows:
1
2
(
u
+
1
)
+
1
6
(
u
−
1
)
−
4
3
(
2
u
+
1
)
I
=
1
2
∫
1
u
+
1
d
u
+
1
6
∫
1
u
−
1
d
u
−
2
3
∫
2
(
2
u
)
+
1
d
u
This can be easily integrated.
I
=
1
2
ln
|
u
+
1
|
+
1
6
ln
|
u
−
1
|
−
2
3
ln
|
2
u
+
1
|
+
C
subst. back ,
u
=
cos
x
I
=
1
2
ln
|
cos
x
+
1
|
+
1
6
ln
|
cos
x
−
1
|
−
2
3
ln
|
2
cos
x
+
1
|
+
C
Hopefully this helps!
Note:
A
,
B
,
C
can obtain by putting
u
=
−
1
,
u
=
1
and
u
=
1
2
into
BLUE EQUATION :
A
(
u
−
1
)
(
2
u
+
1
)
+
B
(
u
+
1
)
(
2
u
+
1
)
+
C
(
u
+
1
)
(
u
−
1
)
=
1
u
=
−
1
⇒
A
(
−
1
−
1
)
(
−
2
+
1
)
+
B
(
0
)
+
C
(
0
)
=
1
⇒
A
(
−
2
)
(
−
1
)
=
1
⇒
A
=
1
2
u
=
1
⇒
A
(
0
)
+
B
(
1
+
1
)
(
2
+
1
)
+
C
(
0
)
=