Question

Given: f'(x)=cscx(cotx-sin2x) what is f(x)?
note: sin2x=2sinxcosx

Answer 1

need a step by step explanation

Answer 2

well this question is about integration

Answer 3

sure will do

Answer 4

yes

Answer 5

im not sure what to do tho lol

Answer 6

not to worry follow my steps you'll get to know

Answer 7

ok

Answer 8

please confirm if i copied your question correctly: \[f'(x) = \csc(x)(\cot(x)-\sin(2x))\]

Answer 9

yup

Answer 10

ok good

Answer 11

ok

Answer 12

my net is a bit slow today i need to format my computer but i sill show you all the steps ok

Answer 13

*bookmark. I am curious to see if mathsmind ever gets to the end of an explaination in a finite amount of time

Answer 14

lol ok

Answer 15

you have been chasing me all day long mr james bond

Answer 16

ok the first step: is simplification

Answer 17

ok.

Answer 18

now you should know by definition what cscx =?

Answer 19

ok recall

Answer 20

\[\csc(x) = \frac{ 1 }{ \sin(x) }\]

Answer 21

1/sin

Answer 22

good .. good

Answer 23

ok so now do i integrate my new equation\'?

Answer 24

Let's push this along a little, shall we?

\[ f'(x) = \csc(x)(\cot(x)−\sin(2x)) = \frac{\cos x}{\sin^2 x} - 2\cos x \]
yes?

Answer 25

\[\int\limits \csc(x)(\cot(x)-\sin(2x))dx\]

Answer 26

no i will make it simpler

Answer 27

\[\int\limits\limits\limits (\csc(x)\cot(x)-\csc(x)\sin(2x))dx\]

Answer 28

\[\int\limits \csc(x)\cot(x) dx - \int\limits \csc(x)\sin(2x) dx\]

Answer 29

now here for sin(2x) i will use the trig identity you provided in the hint which is sin(2x) =2sin(x)cos(x)

Answer 30

and we will use the definition of csc(x)

Answer 31

this yields to

Answer 32

\[\int\limits\limits \csc(x)\cot(x) dx - \int\limits\limits \ \frac{ 1 }{ \sin(x) }2\cos(x)\sin(x) dx\]

Answer 33

You see? The questioner has gone. Are you there, @bmelyk?

Answer 34

look at the 2nd integral

Answer 35

\[\int\limits\limits\limits \csc(x)\cot(x) dx - 2\int\limits\limits\limits \ \cos(x) dx\]

Answer 36

\[\int\limits\limits\limits\limits \csc(x)\cot(x) dx= - \csc(x) + C_1\]

Answer 37

\[- 2\int\limits\limits\limits\limits \ \cos(x) dx=-2\sin(x) +C_2\]

Answer 38

final answer is:

Answer 39

\[=-(\csc(x) + 2\sin(x)) + C\]

Answer 40

i hope this helps sorry my net is frustrating today