Csc4 x - cot* x = CSc?x + cot? x

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