Please for some help with this integration problem.
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
Trying pratial fractions,
When then and
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 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.