integration of cos2x by cos x
Answers
Answer:
−
arctan
(
sin
x
√
cos
2
x
)
+
C
Explanation:
∫
√
cos
2
x
cos
x
⋅
d
x
=
∫
cos
x
⋅
√
cos
2
x
(
cos
x
)
2
⋅
d
x
=
∫
cos
x
⋅
√
1
−
2
(
sin
x
)
2
(
cos
x
)
2
⋅
d
x
=
∫
cos
x
⋅
√
1
−
2
(
sin
x
)
2
1
−
(
sin
x
)
2
⋅
d
x
After using
√
2
⋅
sin
x
=
sin
u
,
√
2
⋅
cos
x
⋅
d
x
=
cos
u
⋅
d
u
and
sin
x
=
sin
u
√
2
transforms, this integral became
∫
(
cos
u
⋅
d
u
√
2
)
⋅
√
1
−
(
sin
u
)
2
1
−
(
sin
u
√
2
)
2
=
∫
√
2
⋅
(
cos
u
)
2
2
−
(
sin
u
)
2
⋅
d
u
=
√
2
⋅
∫
(
cos
u
)
2
1
+
(
cos
u
)
2
⋅
d
u
After using
z
=
tan
u
,
d
u
=
d
z
z
2
+
1
and
(
cos
u
)
2
=
1
z
2
+
1
transforms, it became
=
√
2
⋅
∫
1
z
2
+
1
1
+
1
z
2
+
1
⋅
d
z
z
2
+
1
=
∫
√
2
⋅
d
z
(
z
2
+
1
)
⋅
(
z
2
+
2
)
=
∫
√
2
⋅
d
z
z
2
+
1
−
∫
√
2
⋅
d
z
z
2
+
2
=
√
2
⋅
arctan
z
−
arctan
(
z
√
2
)
+
C
=
√
2
⋅
arctan
(
tan
u
)
−
arctan
(
tan
u
√
2
)
+
C
=
u
√
2
−
arctan
(
tan
u
√
2
)
+
C
After using
√
2
⋅
sin
x
=
sin
u
,
u
=
arcsin
(
√
2
⋅
sin
x
)
and
tan
u
=
√
2
⋅
sin
x
√
1
−
2
(
sin
x
)
2
=
√
2
⋅
sin
x
√
cos
2
x
inverse transforms, I found
∫
√
cos
2
x
cos
x
⋅
d
x
=
√
2
⋅
arcsin
(
√
2
⋅
sin
x
)
−
arctan
(
sin
x
√
cos
2
x
)
+
C
Related questions
Sorry i could not get it in regular form
Step-by-step explanation: