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Math Help - Idefinite integration of sin^2(sinx)+cos^2(cosx)

  1. #1
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    Idefinite integration of sin^2(sinx)+cos^2(cosx)

    Hello. I need help in finding of the indefinite integral of
    \int{Sin^2(sinx)+Cos^2(cosx)} dx from 0 to p/2.

    I've started from reducing squares by  sin(cosx)=(1-Cos(2cosx))/2. the same with cos

    And then I used universal trigonometric substituon. And I got 3 integrals, two of those are really difficult to solve.

    Am I on a right way? Is there any easier way? Because I don't know how to find 1 integral I've got.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Serg777 View Post
    Hello. I need help in finding of the indefinite integral of
    \int{Sin^2(sinx)+Cos^2(cosx)} dx from 0 to p/2.

    I've started from reducing squares by  sin(cosx)=(1-Cos(2cosx))/2. the same with cos

    And then I used universal trigonometric substituon. And I got 3 integrals, two of those are really difficult to solve.

    Am I on a right way? Is there any easier way? Because I don't know how to find 1 integral I've got.
    Well QickMath (which is Mathematica in disquise) does not know a closed form for the indefinite integral, and the Maxima Integrate and Risch Integrate also do not know how to handle this integral.

    CB
    Last edited by CaptainBlack; February 25th 2009 at 01:02 PM.
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  3. #3
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    Try to prove that \int_{0}^{\frac{\pi }{2}}{\big(\sin ^{2}(\sin x)+\cos ^{2}(\cos x)\big)\,dx}=\int_{0}^{\frac{\pi }{2}}{\big(\sin ^{2}(\cos x)+\cos ^{2}(\sin x)\big)\,dx}, thus by adding these we have \int_{0}^{\frac{\pi }{2}}{(1+1)\,dx}=\pi, and divide by two this to get the answer.
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  4. #4
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    very interesting way.I'll try.thank you.
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  5. #5
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    This is a classic example which does tell you: do not attempt to find an antiderivative!

    There's no way in terms of elementary functions, so don't waste your time. Then, we have some properties on given intervals which are useful in cases like this.
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  6. #6
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    Is it appropriate to show it graphically stating that those functions are in same range and that the areas below curves must be the same?
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  7. #7
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    Well, there's an algebraic way of doing it, but yours it's a start.
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  8. #8
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    Ok. I think I know how. Thank you. You have contributed really much to my math education.
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  9. #9
    Member arpitagarwal82's Avatar
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    Buddy as Krizalid mentioned whenever you see a integral, dont jump in to find the anti derivative. At this level we can say that not all functions have indefinite anti derivatives.

    A very simple looking example would be e to the power x^2.
    Try ti integrate this simle function and you will get the point.
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