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Math Help - Integrate 3cos^2(x)

  1. #1
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    Exclamation Integrate 3cos^2(x)

    Hey guys,

    Can you please show me a step by step integration for


    Find the solution for the differential equation :

    3Cos^2(x) , y= Pi , x = Pi/2



    Thank you !
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  2. #2
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    Quote Originally Posted by ZaZu View Post
    Hey guys,

    Can you please show me a step by step integration for


    Find the solution for the differential equation :

    3Cos^2(x) , y= Pi , x = Pi/2



    Thank you !
    Note that \cos (2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 \Rightarrow \cos^2 x = \frac{\cos (2x) + 1}{2}.
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  3. #3
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    Thanks for the reply, but where did the Cos2(x) come from ?!
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  4. #4
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    Quote Originally Posted by ZaZu View Post
    Thanks for the reply, but where did the Cos2(x) come from ?!
    Are you familiar with the double angle formulae, specifically the fact that \cos (2x) = \cos^2 x - \sin^2 x ? That was my starting point.
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  5. #5
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    Yes I do, but I dont understand how using that will affect my answer :S
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  6. #6
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    Quote Originally Posted by ZaZu View Post
    Yes I do, but I dont understand how using that will affect my answer :S
    For goodness sake! I have shown you how to write \cos^2 x in terms of things you should be able to integrate.

    Do you know how to integrate \frac{1}{2} (\cos (2x) + 1) ?
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  7. #7
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    Is it 1/4 Sin2(x) + x ?
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  8. #8
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    Quote Originally Posted by ZaZu View Post
    Is it 1/4 Sin2(x) + x ?
    Only half right. You needed to realise that \frac{1}{2} (\cos (2x) + 1) = \frac{1}{2} \cos (2x) + \frac{1}{2} \, ....
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  9. #9
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    Oh sorry didnt realize the 1/2 is multiplied by the whole thing in brackets

    So its 1/4 Sin2(x) + x/2
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  10. #10
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    Quote Originally Posted by ZaZu View Post
    Oh sorry didnt realize the 1/2 is multiplied by the whole thing in brackets

    So its 1/4 Sin2(x) + x/2
    + C.

    And now your job is to find the value of C using the given data.
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  11. #11
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    Ohhhh okay got it

    So its a rule that I should know, when I have an even power, I should double the angle to get an easier function right ???
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  12. #12
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    ummm, sort of. You do have to use double angle formulae if that's what you mean.

    Think of this:

    \int cos^4 \theta \ d \theta=\int (cos^2 \theta)^2 \ d \theta

    We already know that (i'll show how Mr F gets his answer since you've never seen it before):

    cos^2 \theta +sin^2 \theta=1 and cos^2 \theta - sin^2 \theta= cos 2\theta.

    We require an expression for cos^2 \theta.

    cos^2 \theta +sin^2 \theta=1 \Rightarrow \ sin^2 \theta = 1-cos^2 \theta from the first identity.

    Putting this into the second identity gives:

    2cos^2 \theta -1=cos 2 \theta \Rightarrow \ cos^2 \theta = \frac{cos 2 \theta +1}{2}.

    Going back to our original integral:

    \int cos^4 \theta \ d \theta=\int (cos^2 \theta)^2 \ d \theta = \ \int \left( \frac{cos 2 \theta +1}{2} \right)^2 d \theta =\frac{1}{4} \int (cos 2 \theta +1)^2 d \theta=\frac{1}{4} \int cos^2 2 \theta +2 cos 2 \theta +1 d \theta.

    The big problem now is the cos^2 2 \theta.

    Looking at the integral you posted and what I have done here, can you finish this integral off?

    (Hint: cos^2 (2 \theta)+sin^2 (2 \theta)=1 and cos^2 (2 \theta)- sin^2 (2 \theta)=cos(4 \theta)).
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  13. #13
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    Thanks, Ill get on with it and ill reply when im done
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