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Thread: Integration of sin3x

  1. #1
    Newberry
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    Arrow Integration of sin3x

    i found this for help cos i was having trouble and thought it may help all of you too.

    Consider the expression sin3x. We will use the addition formulae and double angle formulae towrite this in a different form using only terms involving sinx and its powers.

    We begin by thinking of 3x as 2x + x and then using an addition formula:

    sin3x = sin(2x + x)= sin2xcosx + cos2xsinx

    using the first addition formula=
    (2sinxcosx)cosx + (1 − 2sin2x)sinx

    using the double angle formula
    cos2x = 1 − 2sin2x= 2sinxcos2x + sinx − 2sin3x= 2sinx(1 − sin2x) + sinx − 2sin3x

    from the identity cos2x + sin2x = 1= 2sinx − 2sin3x + sinx − 2sin3x= 3sinx − 4sin3x

    We have derived another identity

    sin3x = 3sinx − 4sin3x

    Note that by using these formulae we have written sin3x in terms of sinx (and its powers). You could carry out a similar exercise to write cos3x in terms of cosx.

    it came from this website
    http://72.14.203.104/search?q=cache:...u&ct=clnk&cd=1

  2. #2
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    Re: Integration of sin3x

    Thx for this, but this is common practise

    And i dont think you mean the integration of sin3x, but rather the integration of a trig function that is powered to a certain degree...

  3. #3
    Newberry
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    Question Re: Integration of sin3x

    k thanx lol. can u please do me a huge favour and tell me the answer you get for
    the integration of sin^3 x between pi/6 and zero. i keep getting the answer around the wrong way?!?

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    Re: Integration of sin3x

    I got (3.sqrt3)/8 - 2/3

  5. #5
    Newberry
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    Re: Integration of sin3x

    i did too, but the answer is around the other way ie 2/3 - (3.sqrt3)/8
    it is question 7b in the maths in focus text book

  6. #6
    icycloud
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    Re: Integration of sin3x

    You stated the limits as between pi/6 and zero, which implies pi/6 --> 0, and the answer is what Riviet got.

    If however the limits are 0-->pi/6, then the answer is the other way around.

    Because I = ∫(a-->b) f(x) dx = -∫(b-->a) f(x) dx

    Hope this helps.

  7. #7
    Newberry
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    Re: Integration of sin3x

    sorry, i mean question 7b out of exercise 5.11

  8. #8
    Newberry
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    Re: Integration of sin3x

    yeah thanx

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    Re: Integration of sin3x

    Icycloud read my mind. XD

    Basically, if it was:
    pi/6
    ∫ then we get our answer but
    0

    0
    -∫ it would be reversed, by icycloud's reasoning.
    pi/6

    Anyway, it's all cool now.
    Last edited by Riviet; 21 Mar 2006 at 6:53 AM.

  10. #10
    Newberry
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    Smile Re: Integration of sin3x

    actually we are all wrong. it was a simple error, when you integrate sinx, you get -cosx, thats where i went wrong, i integrated sinx to equal cosx.

  11. #11
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    Re: Integration of sin3x

    Hang on a minute...so you actually do mean the integration of sin3x, and not sin^3 x

    Why are you going through all that mumbo jumbo for?

    The integration of sin3x is just -(cos3x)/3



    And if you were to find the integration of sin^3 x, it will be easier to use substitution
    sin^3(x) = (1 - cos^2 (x) ) . sinx

    Then let u = cos(x), du = -sin(x)dx

    Then you get:
    -∫1-u^2 du
    -u + (u^3)/3

    Sub u back in, then sub in original borders...or change the borders first, then sub new borders into that equation
    Last edited by _ShiFTy_; 21 Mar 2006 at 5:47 PM.

  12. #12
    Newberry
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    Re: Integration of sin3x

    no, i do mean sin^3x. its just that when i integrated sin after rearanging sin^3x to equal 1/4(3sinx-sin3x) i got cos instead of negative cos for the integration of sin. i didnt want to use a substitution either because that still means that you have to expand the answer, and i find this way to be much easier.
    anyways, i'm past that question, but have another one if you wouldnt mind helping?!?

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    Re: Integration of sin3x

    The world is waiting for your question

  14. #14
    Newberry
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    Red face Re: Integration of sin3x

    thanx. integration of cos^3x. im not sure where to start with it

  15. #15
    Executive Member rama_v's Avatar
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    Re: Integration of sin3x

    Quote Originally Posted by Newberry
    thanx. integration of cos^3x. im not sure where to start with it
    (Cos3@) can be integrated by using the complex identity Cos@ = e@ + [e-i@][/ 2

    where ei@=Cos@ + iSin@

    let z = ei@
    1/z = e-i@

    ((z + (1/z))/2)3 = Cos3@
    Go from there

  16. #16
    Newberry
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    Question Re: Integration of sin3x

    i dont do extension 2, and we havent done "e" yet either - any other way to do it?

  17. #17
    Executive Member rama_v's Avatar
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    Re: Integration of sin3x

    Quote Originally Posted by Newberry
    i dont do extension 2, and we havent done "e" yet either - any other way to do it?
    Oh Sorry..Umm I'm not sure of another way...

  18. #18
    Newberry
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    Wink Re: Integration of sin3x

    'tis ok, thanx anyway

  19. #19
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    Re: Integration of sin3x

    Use subsitution, its a lot easier than you think

    cos3@ = (1 - sin2@) x cos@

    Therefore the:
    ∫cos3@ d@
    =∫ (1 - sin2@) x cos@ d@

    Let u = sin@
    so du = cos@.d@
    Substituting these in, you get the

    ∫ (1 - u2 )du ....... *now this is very easy to integrate
    u - u3/3

    By substituting "u" back in, we get
    sin@ - (sin3@)/3


  20. #20
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    Re: Integration of sin3x

    there is another way...

    realise that cos3x = cos(2x+x) = cos2xcosx - sin2xsinx
    = (2cos^2x - 1)cosx - (2sinxcosx)sinx
    = 2cos^3x - cosx - 2(1-cos^2x)(cosx)
    =4cos^3x - 3cosx

    SO...since cos3x = 4cos^3x - 3cosx

    THEN ==> cos^3x = 1/4[cos3x +3cosx], which you can integrate.
    B Comm/LLB @ USyd (V)


    Yeah, I feel old...as an undergrad. lol.

  21. #21
    Newberry
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    Talking Re: Integration of sin3x

    thanks, that was what i was trying to do, but i wasnt sure if you could multiply 2cos^2x by cosx, but apparently you can. thnx again

  22. #22
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    Re: Integration of sin3x

    a pleasure
    B Comm/LLB @ USyd (V)


    Yeah, I feel old...as an undergrad. lol.

  23. #23
    YBK
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    Re: Integration of sin3x

    Quote Originally Posted by Newberry
    actually we are all wrong. it was a simple error, when you integrate sinx, you get -cosx, thats where i went wrong, i integrated sinx to equal cosx.
    integral of sinx x is cosx

    but integral of cos x is -sinx

  24. #24
    icycloud
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    Re: Integration of sin3x

    Quote Originally Posted by YBK
    integral of sinx x is cosx

    but integral of cos x is -sinx
    It's the other way around YBK.

  25. #25
    YBK
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    Re: Integration of sin3x

    Quote Originally Posted by icycloud
    It's the other way around YBK.
    OOPS!
    We only did differentiation at school... that's what I was thinking of

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