1) the integral cos² X dx is equal to
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Answer:
Explanation:
e have:
I
=
x
cos
2
(
x
)
d
x
=
∫
x
sec
2
(
x
)
d
x
We will use integration by parts, which takes the form
∫
u
d
v
=
u
v
−
∫
v
d
u
. For the given integral
∫
x
sec
2
(
x
)
d
x
, let:
{
u
=
x
d
u
d
x
=
1
d
u
=
d
x
d
v
=
sec
2
(
x
)
d
x
∫
d
v
=
∫
sec
2
(
x
)
d
x
v
=
tan
(
x
)
Thus:
I
=
x
tan
(
x
)
−
∫
tan
(
x
)
d
x
I
=
x
tan
(
x
)
+
∫
−
sin
(
x
)
cos
(
x
)
d
x
Letting
y
=
cos
(
x
)
such that
d
y
=
−
sin
(
x
)
d
x
:
I
=
x
tan
(
x
)
+
∫
d
y
y
I
=
x
tan
(
x
)
+
ln
(
|
y
|
)
I
=
x
tan
(
x
)
+
ln
(
|
cos
(
x
)
|
)
+
C
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