integral sin2x/(sin²x+1)(sin²x+3)
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Step-by-step explanation:
Reminder
cos
2
x
=
1
−
2
sin
2
x
sin
2
x
=
1
2
(
1
−
cos
2
x
)
Integration by parts
∫
u
v
'
=
u
v
−
∫
u
'
v
The integral is
I
=
∫
x
2
sin
2
x
d
x
=
1
2
∫
x
2
(
1
−
cos
2
x
)
d
x
=
1
2
∫
x
2
−
1
2
∫
x
2
cos
2
x
d
x
=
1
6
x
3
−
1
2
∫
x
2
cos
2
x
d
x
Perform the second integral by parts
u
=
x
2
,
⇒
,
u
'
=
2
x
v
'
=
cos
2
x
,
⇒
,
v
=
1
2
sin
2
x
So,
∫
x
2
cos
2
x
d
x
=
x
2
2
sin
2
x
−
∫
x
sin
2
x
d
x
Calculate the third intehral by parts
u
=
x
,
⇒
,
u
'
=
1
v
'
=
sin
2
x
,
⇒
,
v
=
−
1
2
cos
2
x
So,
∫
x
sin
2
x
d
x
=
−
x
2
cos
2
x
+
1
2
∫
cos
2
x
d
x
=
−
x
2
cos
2
x
+
1
4
sin
2
x
Putting it all together
I
=
1
6
x
3
−
x
2
4
sin
2
x
−
x
4
cos
2
x
−
1
8
sin
2
x
+
C
Answered by
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