# Thread: Help on a hard integration problem

1. ## 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

2. ## Re: Help on a hard integration problem

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

$\displaystyle \sin \theta=\frac{t}{\sqrt{1+t^2}},\quad \cos \theta=\frac{1}{\sqrt{1+t^2}},\;d\theta=\frac{dt}{ 1+t^2}$

3. ## Re: Help on a hard integration problem

Originally Posted by Olbert
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: $\displaystyle \int\frac{\cos^2(\theta)}{A^2-\sin^2(\theta)} d\theta$

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

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

But Fernando's suggestion is, no doubt, better.

4. ## 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