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Math Help - Help on a hard integration problem

  1. #1
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    Help on a hard integration problem

    Hi - this is my first post on this forum so please be nice.

    I have been thinking about a problem my university physics lecturer proposed and explained in class:

    Where would an object be pulled by gravity if it was on the inside of a hollow sphere with the same size and mass as the earth?

    This was years ago and I forgot the answer, but I set about to answer it mathematically using the various laws of physics I remember.

    Anyways, long story I am stuck with a fairly ugly integral. A quick search of the standard integrals on wiki heralded no solution.

    I need to solve an integral in the form of:

    Integrate: (cos^2(theta)/(A^2-sin^2(theta)) d(theta)

    I have no idea how to write that properly but if I was to say it to someone I would say :Integral of cos squared on 'a' squared minus sine squared.

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Help on a hard integration problem

    Quote Originally Posted by Olbert View Post
    Integrate: (cos^2(theta)/(A^2-sin^2(theta)) d(theta)
    There is a standard way using the substitution t=\tan \theta . In such a case

    \sin \theta=\frac{t}{\sqrt{1+t^2}},\quad \cos \theta=\frac{1}{\sqrt{1+t^2}},\;d\theta=\frac{dt}{  1+t^2}
    Last edited by FernandoRevilla; November 12th 2011 at 02:28 AM.
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  3. #3
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    Re: Help on a hard integration problem

    Quote Originally Posted by Olbert View Post
    Hi - this is my first post on this forum so please be nice.

    I have been thinking about a problem my university physics lecturer proposed and explained in class:

    Where would an object be pulled by gravity if it was on the inside of a hollow sphere with the same size and mass as the earth?

    This was years ago and I forgot the answer, but I set about to answer it mathematically using the various laws of physics I remember.

    Anyways, long story I am stuck with a fairly ugly integral. A quick search of the standard integrals on wiki heralded no solution.

    I need to solve an integral in the form of:

    Integrate: (cos^2(theta)/(A^2-sin^2(theta)) d(theta)

    I have no idea how to write that properly but if I was to say it to someone I would say :Integral of cos squared on 'a' squared minus sine squared.

    Thanks
    So, you need to integrate: \int\frac{\cos^2(\theta)}{A^2-\sin^2(\theta)} d\theta

    \frac{\cos^2(\theta)}{A^2-\sin^2(\theta)}=\frac{1}{\displaystyle \frac{A^2}{\cos^2(\theta)}-\frac{\sin^2(\theta)}{\cos^2(\theta)}}=\frac{1}{A^  2\sec^2(\theta)-\tan^2(\theta)}

    Also, sec^2(x) - tan^2(x) = 1 .

    But Fernando's suggestion is, no doubt, better.
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  4. #4
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    Re: Help on a hard integration problem

    Wow, I haven't used the relationship between t and theta since highschool. I completely forgot. Thanks heaps
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