Is this possible to integrate?

[jason17349]

I haven't had any luck with mathcad and was wondering if this was possible to integrate...

$$\int_{0}^{2\pi} \frac {x+r*cos(\theta)}{(x^2+2r*x*cos(\theta)+r^2)^\frac {3}{2}} d\theta$$

 

[daveb]

It's certainly integrable...now whether you can solve it by non-numerical methods may be another matter entirely.

 

[jason17349]

Does anybody have any suggestions on how to go about integrating this... ? Or maybe explain why mathcad isn't able to integrate this..

 

[StatusX]

Do you have to integrate over theta first? If you could integrate over x first, you could use substitution. That's usually the easiest way to do integrals like these.

 

[jason17349]

Sorry, I should have said x and r are real non-negative constants..

 

[StatusX]

Even so, you can pretend it's a variable. Then, integrate with respect to x to get some function F, so that your integral becomes:

$$\int \left( \frac{\partial}{\partial x} F \right) d\theta = \frac{\partial}{\partial x} \left( \int F d\theta \right)$$

I don't know if that helps, but it's a cool trick.

 

[benorin]

Gee, that looks an awful lot like Poisson's Integral.

 

[benorin]

Maple gives an answer involving Elliptic integrals of the first and second kind, but in terms of those integrals, the answer is not that bad (though long enough for me not to post it.)