Integral of Cosec2x

My mate just asked me how to do this question below. She is using a very complicated method, substituting sinx for tanxcosx!

I'm sure they're must be an easier way! Any idea, I'm too tired to think straight, but no doubt I'll dream about it all night like usual and wake up at 3am going "I've got it!!!".

Thanks:

Integral of Cosec2x

Reply 1

diablo2004

is that cosec 2x or cosec²x?

Reply 2

Hoofbeat

OP

14

diablo2004

is that cosec 2x or cosec²x?

cosec 2x

Reply 3

kikzen

14

Hoofbeat

My mate just asked me how to do this question below. She is using a very complicated method, substituting sinx for tanxcosx!

I'm sure they're must be an easier way! Any idea, I'm too tired to think straight, but no doubt I'll dream about it all night like usual and wake up at 3am going "I've got it!!!".

Thanks:

Integral of Cosec2x

ive seen the integration of cosec x. they used something complicated and i didnt really understand why...

Reply 4

diablo2004

cosec 2x = -(-cosex 2x cot 2x -cosec² 2x)/(cosec 2x - cot 2x)

= derivative/function => integral = log (function)

=> integral (cosec 2x) = -1/2 log [A(cosec 2x - cot 2x)] = log [A(cosec2x - cot2x)]^-1/2

Reply 5

Bezza

2

Hoofbeat

Integral of Cosec2x

The answer is 1/2*ln(cosec(2x) - cot(2x)), not sure why though! I just used http://www.calc101.com

Reply 6

mikesgt2

You can use the method suggested by diablo2004, or the other way to do it is to substitute t=tan(x) to obtain the integral as ln|tan(x)|.

Reply 7

Hoofbeat

OP

14

diablo2004

cosec 2x = -(-cosex 2x cot 2x -cosec² 2x)/(cosec 2x - cot 2x)

= derivative/function => integral = log (function)

=> integral (cosec 2x) = -1/2 log [A(cosec 2x - cot 2x)] = log [A(cosec2x - cot2x)]^-1/2

How did you get the first line?!

Reply 8

Fermat

8

Hoofbeat

How did you get the first line?!

cosec 2x = -(-cosex 2x cot 2x -cosec² 2x)/(cosec 2x - cot 2x)

He multiplied the cosec 2x by 1= (cosec 2x - cot 2x)/(cosec 2x - cot 2x)

see here for explanation.

I did it the same way as mikesgt2 with the t=tan(x) substitution.

Reply 9

Silly Sally

I got the final ans. as 0.5ln(tanx)

The way i did it:

cosecx = 1/(sin2x)
= 1/(2sinxcosx)

Then used sinx = tanxcosx and substitued this into eqn:

= 1/(2cosxtanxcosx)
= 1/(2tanx (cosx)^2))
= 0.5 ((secx)^2/tanx)

To find integral use substitution method as differential of tanx = (secx)^2

so final ans

integral (cosec2x) = 0.5ln(tanx)

I don't know whether that is the right ans - but does that make sense?!?

Reply 10

Fermat

8

Silly Sally

I got the final ans. as 0.5ln(tanx)

The way i did it:

cosecx = 1/(sin2x)
= 1/(2sinxcosx)

Then used sinx = tanxcosx and substitued this into eqn:

= 1/(2cosxtanxcosx)
= 1/(2tanx (cosx)^2))
= 0.5 ((secx)^2/tanx)

To find integral use substitution method as differential of tanx = (secx)^2

so final ans

integral (cosec2x) = 0.5ln(tanx)

I don't know whether that is the right ans - but does that make sense?!?

Yes that's (a) right answer.
I got that and so did mikesgt2.

The expression (cosec 2x - cot 2x) expands to tan(x), so thay are all the same answer!

Reply 11

Silly Sally

Thanks!!! - At least i know that someone agrees with me :smile:

Reply 12

Hoofbeat

OP

14

Silly Sally

Thanks!!! - At least i know that someone agrees with me :smile:

Hey Sally! I was in the jucazzi at the gym this afternoon with my mum when I figured it out! I used your method (I swear when you were doing it last night it on msn it was a lot more complicated!!!). My mum said I was interesting to watch as I was just sitting in the jucazzi looking around but looking as though I was working something out and then I went "Yes! I've done it!" lol. She laughed at me and said I really should stop thinking about Maths all the time - especially on my day-off from revision! Couldn't help it though! :tongue:

Chat to you later on msn probably!
Chloé

Reply 13

diablo2004

Silly Sally

I got the final ans. as 0.5ln(tanx)

The way i did it:

cosecx = 1/(sin2x)
= 1/(2sinxcosx)

Then used sinx = tanxcosx and substitued this into eqn:

= 1/(2cosxtanxcosx)
= 1/(2tanx (cosx)^2))
= 0.5 ((secx)^2/tanx)

To find integral use substitution method as differential of tanx = (secx)^2

so final ans

integral (cosec2x) = 0.5ln(tanx)

I don't know whether that is the right ans - but does that make sense?!?

Yeah, that's probably a better method than mine, if you don't know what to do for the first line.

Reply 14

Hoofbeat

OP

14

Hoofbeat

Hey Sally! I was in the jucazzi at the gym this afternoon with my mum when I figured it out! I used your method (I swear when you were doing it last night it on msn it was a lot more complicated!!!). My mum said I was interesting to watch as I was just sitting in the jucazzi looking around but looking as though I was working something out and then I went "Yes! I've done it!" lol. She laughed at me and said I really should stop thinking about Maths all the time - especially on my day-off from revision! Couldn't help it though! :tongue:

Chat to you later on msn probably!
Chloé

One thing though Sally. Once you get 0.5(sec^2x)/tanx you don't actually need to use a proper substitution method as this is f'(x)/f(x) = ln[f(x)] + c.

Reply 15

Barny

17

Hoofbeat

Hey Sally! I was in the jucazzi at the gym this afternoon with my mum when I figured it out! I used your method (I swear when you were doing it last night it on msn it was a lot more complicated!!!). My mum said I was interesting to watch as I was just sitting in the jucazzi looking around but looking as though I was working something out and then I went "Yes! I've done it!" lol. She laughed at me and said I really should stop thinking about Maths all the time - especially on my day-off from revision! Couldn't help it though! :tongue:

Chat to you later on msn probably!
Chloé

Wow, nice little story, shame its hard to believe a word of it. For a start, you wouldnt be able to work it out in your head, secondly, you wouldnt shout out 'Ive done it' like some sort of rip off from the stephen hawking programme on BBC2 the other night and thirdly, that is sickeningly geeky.

Reply 16

4Ed

15

imasillynarb

Wow, nice little story, shame its hard to believe a word of it. For a start, you wouldnt be able to work it out in your head, secondly, you wouldnt shout out 'Ive done it' like some sort of rip off from the stephen hawking programme on BBC2 the other night and thirdly, that is sickeningly geeky.

you're missing the point here...

OMG HOOFBEAT YOU HAVE A JACUZZI!!! :biggrin:

Reply 17

Barny

17

4Ed

you're missing the point here...

OMG HOOFBEAT YOU HAVE A JACUZZI!!! :biggrin:

In the jacuzzi at the gym, not hers.

Reply 18

Hoofbeat

OP

14

Yeah it was at a local Gym. And Ok, so i didn't excatly shout it but I did say quite loudly "I've done it!". Trust me, it's the sort of thing I would do!

Reply 19

Barny

17

Hoofbeat

Yeah it was at a local Gym. And Ok, so i didn't excatly shout it but I did say quite loudly "I've done it!". Trust me, it's the sort of thing I would do!

Maybe you should consider getting a life then..

Integral of Cosec2x

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