simple 3u..plz help (1 Viewer)

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um...when d/dx sin^n x (where n=integer)

i was told to make it d/dx (sinx)^n

so ok, fair enough
so how come when integrating sin^2 x. i cannot integrate it as (sinx)^2 ?? but rather 1-cos2x...

 

[Constip8edSkunk]

sin^2 x is just another way of writing (sinx)^2 its the same thing

either way you'll have to change to 1-cos2x

u cant integrate straight away coz I f'(x)[f(x)]^n=[[F(x)]^(n+1)]/n+1 and no cosx (derivative of sinx) is there

 

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Originally posted by Constip8edSkunk
sin^2 x is just another way of writing (sinx)^2

yea, but if i were to do d/dx sin^2 x. all i needed to do is make it d/dx (sinx)^2 and do the product(or chain, whatever which one) rule to get the anwer..i dun need to change it to 1-cos2x

Originally posted by Constip8edSkunk
u cant integrate straight away coz I f'(x)[f(x)]^n=[[F(x)]^n+1]/n+1 and no cosx (derivative of sinx) is there

ok, i am even more lost now...

 

[freaking_out]

Originally posted by ...
um...when d/dx sin^n x (where n=integer)

i was told to make it d/dx (sinx)^n

so ok, fair enough
so how come when integrating sin^2 x. i cannot integrate it as (sinx)^2 ?? but rather 1-cos2x...

well in the first case u can make it (sinx)^n, but the dervative turns out to be: n(sinx)^(n-1)cosx. so only when u integrate this, would u get the answer to be in the form sinⁿx. hence u can c how u cannot just integrate.

if u want, why don't u try it, and c what u get, and then differentiate it....u'll c how the (sinx)^2 wouldn't work.

 

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hmm, yea, i kinda understand...

hmm, will we get told how to do sin^3 x in 3u then?? or we can use sub method?

 

[Constip8edSkunk]

im not actually sure what you are asking

I sin^2x dx = x/2 - [sin2x]/4 +c

if you can get that straight off from looking at sin^2 x then thats very good, changing it to 1-cos2x makes it simpler

if f(x) =sinx
then dsin^nx/dx = n[f(x)]^n-1 * f'(x)

so reversing the process by integrating,

I {n[f(x)]^n-1 * f'(x)} = [f(x)]^n + c

or in sinx 's case, I [ nsin^n-1xcosx] = sin^n x + c

 

[freaking_out]

Originally posted by Constip8edSkunk
....so reversing the process by integrating,

I {n[f(x)]^n-1 * f'(x)} = [f(x)]^n + c

or in sinx 's case, I [ nsin^n-1xcosx] = sin^n x + c

hey, came down with all these notations man, he's just a innocent 3u kid.

 

[freaking_out]

Originally posted by ...
hmm, yea, i kinda understand...

hmm, will we get told how to do sin^3 x in 3u then?? or we can use sub method?

yeah, for 3u if u r prolly never gonna be asked to integrate it on its own, but rather u might be asked a substitution integration question such as to integrate: sin³xcosx.....in where u would let u=sinx.

 

[underthesun]

See:

let p = sin(x)

dp/dx = cos(x)

let y = pⁿ = sinⁿ(x)

dy/dp = np^n-1



Now how do we get dy/dx? We multiply dy/du with dp/dx, and you'll get

dy/dx = np^n-1cos(x) = nsin^n-1(x)cos(x)

That's differentiation by substitution for you. From the above result:

when y = sin³(x), dy/dx = 3sin²(x)cos(x)

This means, when you integrate 3sin²(x)cos(x), you'll get sin³(x).

This means, when you integrate 3sin²(x) only, you won't get sin³(x).

edit : I just realised that i was basically saying what freaking did..

 

[Constip8edSkunk]

Originally posted by freaking_out
hey, came down with all these notations man, he's just a innocent 3u kid.

haha... well i was just trying to be general



As for sin^3 x, im not sure that it wont be asked as it only involves trigs transformation, ie sin^3x = sinx(1-cos^2x) then integrate sinx and sinxcos^2x separately


edit: and yeah if they want you to use substitution in 3 Unit, they'll tell u what to sub in.

 

[freaking_out]

Originally posted by underthesun
....edit : I just realised that i was basically saying what freaking did..

lol, i feel sorry for "..." us ext.2'ers speak a completely different language after all.

 

[ND]

Originally posted by ...
hmm, yea, i kinda understand...

hmm, will we get told how to do sin^3 x in 3u then?? or we can use sub method?

For that, use the differential method, check OLDMAN's site for a good explanation.

 

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Originally posted by ND
For that, use the differential method, check OLDMAN's site for a good explanation.

that looks more like on of those band1 english essays...have no idea what he is on about..lol..


btw, i am doing '99 3u HSC paper and q3)d)ii)

its integrating, but looking at worked solution, all of a sudden half way throug the dude added his own substition criteria..(like he made up u=....)

are we allow to do that? if so, what should i put in..

 

[Constip8edSkunk]

Yes... you can even use 4 unit methods in 3 unit tests or so im told, but in 3 unit, if theres a substitution necessary, they'll specify it.

 

[freaking_out]

Originally posted by ...
that looks more like on of those band1 english essays...have no idea what he is on about..lol..

but its a band 6 4u response.

 

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umm...d)ii)

let me scan what i have got to..

 

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here...

 

[freaking_out]

hmmm....in that case u do need 2 know how to choose substitution.

 

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exactly..

so what decision is needed?

 

[ND]

After doing part i), i'm sure it'll become pretty obvious.