Answer:
Step-by-step explanation:
When approaching the integral,
∫
csc
4
(
x
)
cot
6
(
x
)
d
x
, it is helpful to ask about derivatives and integrals of the various functions we see.
d
d
x
(
csc
x
)
=
−
csc
x
cot
x
so perhaps we could split off one of each and rewrite using only
csc
x
. We know that there is a relationship, but it involves squares, not the 3rd and 5th power we have left after separating
csc
x
cot
x
. We'll keep it in mind if we don't get a better idea.
d
d
x
(
cot
x
)
=
−
csc
2
x
. And if we split off a
csc
2
x
, we will have
csc
2
x
remaining and we know that we can rewrite that using
cot
x
, so we'll try that. (with substitution
u
=
cot
(
x
)
(With experience and practice, this reasoning takes place very fast and we know this will work. As students, we have to try something and see if it works.)
∫
csc
4
(
x
)
cot
6
(
x
)
d
x
=
∫
csc
2
(
x
)
cot
6
(
x
)
csc
2
(
x
)
d
x
=
∫
(
cot
2
(
x
)
+
1
)
cot
6
(
x
)
csc
2
(
x
)
d
x
=
∫
(
cot
8
(
x
)
+
cot
6
(
x
)
)
csc
2
x
d
x
=
∫
(
u
8
+
u
6
)
(
−
d
u
)
(
u
=
cot
(
x
)
)
=
−
1
9
u
9
−
1
7
u
7
+
C
=
−
1
9
cot
9
(
x
)
−
1
7
cot
7
(
x
)
+
C
