Please for some help with this integration problem.
i.e.
integral ((a x^4 + b x^2 + c)^–½) dx
Thanks for the rationalize approach. I will look into that.
The problem is based on predicting the rotation of a solid body after any given time. As far as I know there is no yet found solution. The integration is done numerically on a computer. I am desparately trying to get a theoretical formula. Alas, I am not that great with integration; I am at GCE A-level.
The equation is actually correctly stated.
Ok, I will continue with the computer method.Originally Posted by ThePerfectHacker
I actually broke down the numerical method into this itegration and a matrix inverse. I expect that this approach have been attempted before, and hoped against.
Thanks for your expertise.
I have a broken down the original equation using partial fractions
Using the roots of a quadratic equation, let
and
Therefore becomes
Trying pratial fractions,
Therefore
When then and
Likewise
So the original becomes
Expanding to avoid imaginary values
The new form is
How can this proceed further? I noticed that it does not fall exactly in the arcsin value for the integral
The equation was not correct due to the ± issue of the square root. I will try to re-formulate the equation keeping the sign in mind. Consequently, the result I gave in post one would be rather unsolvable.
I also looked at Elliptic Integral and Jacobian Elliptic Functions. I could not follow Closed Form.
You are right: This integral form is done in numerical methods.
I don't know if your still working on it, but maybe this may help The Integral Calculator
I think that I finally understand the closed form and the ellipsoid concept
Firstly, the first equation is not of a 'function' therefore the area under the graph would not be closed, and to explain it more vividly: the eqution is that of an ellipse!
EDIT: I think that the arctan integral is only valid for a range of x<1
Well, I did the equation for this over, if any is interested in helping. I am putting it in the Advance Topics section.