integration of x/(1+sinx) dx
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Answer:
Step-by-step explanation:
x
(
tan
x
−
sec
x
)
+
ln
∣
∣
∣
sec
x
+
tan
x
sec
x
∣
∣
∣
+
c
Explanation:
I
=
∫
x
1
+
sin
x
d
x
=
∫
x
(
1
−
sin
x
)
1
−
sin
2
x
d
x
=
∫
x
(
1
−
sin
x
)
cos
2
x
d
x
⇒
I
=
∫
x
sec
2
x
d
x
−
∫
x
sec
x
tan
x
d
x
Integration by parts,we get
I
=
[
x
tan
x
−
∫
1
⋅
tan
x
d
x
]
−
[
x
⋅
sec
x
−
∫
1
⋅
sec
x
d
x
]
I
=
x
tan
x
−
ln
|
sec
x
|
−
x
sec
x
+
ln
|
sec
x
+
tan
x
|
+
c
=
x
tan
x
−
x
sec
x
+
ln
|
sec
x
+
tan
x
|
−
ln
|
sec
x
|
+
c
=
x
(
tan
x
−
sec
x
)
+
ln
∣
∣
∣
sec
x
+
tan
x
sec
x
∣
∣
∣
+
c
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