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How would you integrate cos^3 x? watch


    Being stupid........how would you integrate cos^3 x ?

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    cheers
    rosie
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    (Original post by crana)
    Being stupid........how would you integrate cos^3 x ?

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    cheers
    rosie
    Can you split it up into (cosx)(cos²x), and then use a trig identity for (cos²x) and use integration by parts?

    BTW I can't be bothered doing it :rolleyes:
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    (Original post by crana)
    Being stupid........how would you integrate cos^3 x ?

    rep for help
    cheers
    rosie
    cos^3 x = cos^2 x.cos x

    Use cos^2 x = 1 - sin^2 x

    (1 - sin^2 x)cos x = cos x - sin^2 xcos x

    Integrate the second bit using a substitution, u = sin x (so du = cos x dx).

    (Original post by Pixelfairy #1)
    Can you split it up into (cosx)(cos²x), and then use a trig identity for (cos²x) and use integration by parts?

    BTW I can't be bothered doing it :rolleyes:
    cheers - my friend just suggested that too.
    i dont thinnk you actually need to integrate by parts? but i could be wrong

    rsoei
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    (Original post by crana)
    cheers - my friend just suggested that too.
    i dont thinnk you actually need to integrate by parts? but i could be wrong

    rsoei
    See above, you can use substitution.

    i used the "what would you differentiate to get this?" method

    Have hit some problems integrating sin2xsin3x though... have got a hideous expression most probably wrong......ideas?
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    (Original post by crana)
    i used the "what would you differentiate to get this?" method

    Have hit some problems integrating sin2xsin3x though... have got a hideous expression most probably wrong......ideas?
    2sinAsinB = cos(A-B) - cos(A+B)

    sin2xsin3x = [cos(2x-3x) -cos(2x+3x)]/2

    sin2xsin3x = [cos(-x) - cos(5x)]/2

    Integrate to get

    [-sin-x - sin5x/5]/2
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    (Original post by crana)
    i used the "what would you differentiate to get this?" method

    Have hit some problems integrating sin2xsin3x though... have got a hideous expression most probably wrong......ideas?
    You can use the factor formulae
    cos a - cos b = -2sin((a+b)/2)sin((a-b)/2)

    Make a = 5x, b = x.
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    (Original post by crana)
    i dont thinnk you actually need to integrate by parts? but i could be wrong.
    Just wondering, couldnt we just do (cosx)^3, then do simple chain rule?

    Bah this is all beyond me anyway.
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    (Original post by Spy_Lord)
    Just wondering, couldnt we just do (cosx)^3, then do simple chain rule?

    Bah this is all beyond me anyway.
    Chain rule is differentiation, not integration
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    (Original post by Spy_Lord)
    Just wondering, couldnt we just do (cosx)^3, then do simple chain rule?

    Bah this is all beyond me anyway.
    chain rule is for defferinating not intergration, so no. AFAIK
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    (Original post by chud)
    chain rule is for defferinating not intergration, so no. AFAIK
    I mean doing the opposite of chain rule . . strange because I recall having done it frequently. What I mean is add 1 to the indice, divide by the indice multiplied by the derivative of what's in the brackets:

    Int(cosx)^3
    =[(cosx)^4]/(4sinx) + c

    I realise though I could very well be wrong as I am only doing Add Maths right now in Year 11, which is a notch below full A Level.

    Actually, after reading some of the replies, I think perhaps opposite of chain rule cannot be done because you want expression without c. Oh well :rolleyes: .
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    (Original post by Spy_Lord)
    I mean doing the opposite of chain rule . . strange because I recall having done it frequently. What I mean is add 1 to the indice, divide by the indice multiplied by the derivative of what's in the brackets:

    Int(cosx)^3
    =[(cosx)^4]/(4sinx) + c

    I realise though I could very well be wrong as I am only doing Add Maths right now in Year 11, which is a notch below full A Level.

    Actually, after reading some of the replies, I think perhaps opposite of chain rule cannot be done because you want expression without c. Oh well :rolleyes: .
    You can't do that (but I can't remember why). aybe someone with a bigger brain will explain.

    MB
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    (Original post by musicboy)
    You can't do that (but I can't remember why). aybe someone with a bigger brain will explain.

    MB
    Only applicable if the thing in the brackets is a linear and simple term, eg 4x, you can't apply with trig or polynomials greater than 1.
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    (Original post by Bhaal85)
    Only applicable if the thing in the brackets is a linear and simple term, eg 4x, you can't apply with trig or polynomials greater than 1.
    Hey thanks for that, I had no idea. No wonder I couldn't recall doing such a question with trig.
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    (Original post by Bhaal85)
    Only applicable if the thing in the brackets is a linear and simple term, eg 4x, you can't apply with trig or polynomials greater than 1.
    I know the limitations, I was requesting an explanation of them.

    MB
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    Using my very limited Mathematic knowledge, I would just integrate cosx three times and multiply!
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    (Original post by tomhitchings)
    Using my very limited Mathematic knowledge, I would just integrate cosx three times and multiply!
    lol. But its a product of a product of a product.
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    u=sinx

    du/dx=cos x

    du/cosx=dx

    int cos^3(x) dx = int cos^2(x) du = int 1 - sin^2(x) du
    =int 1 - u^2 du

    = u - u^3/3

    = sinx -(sin^3(x)/3)
 
 
 
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