# Need help evaluating flux integral!

I have constructed a formula to represent the neutron flux from a disc emitter through an aperture. I have got it down to a double integral of a form that I can't see how to evaluate. Mathematica crashes on this and I am not ready yet to give up and go to numerical solution. Does anyone know a good technique for evaluating integrals of this type?

Int dTheta*Sin(2*Theta)*Sqrt[A^2-(r-D*Tan(Theta))^2]

It is a product of a trig function and then a sqrt with a trig function inside. Integration by parts doesn't improve the radical and I'm looking for analytic solution if possible...any ideas?

fzero
Homework Helper
Gold Member
Trig identities and operations like completing the square inside the square root will put that in the form

$$\int d\theta \sin\theta \sqrt{\beta \cos^2\theta -1},$$

which can be done in closed form by a simple substitution.

Trig identities and operations like completing the square inside the square root will put that in the form

$$\int d\theta \sin\theta \sqrt{\beta \cos^2\theta -1},$$

which can be done in closed form by a simple substitution.

The integrand is of the form

$$\int d\theta \sin2\theta \sqrt{\alpha^2 - (r-\beta\tan\theta)^2}$$

Completing the square will simplify the radical, but the radicand becomes negative in this case:

$$\int d\theta \sin2\theta \sqrt{-\gamma-u^2}$$ where $$u=\tan\theta-r/\beta$$ and $$\gamma = (\alpha^2-2r^2)/\beta^2$$

Our integrating variable $$du/d\theta$$ is then equal to $$1+\tan\theta^2 = 1/\cos^2\theta$$

After rewriting the differential coefficient in terms of u, the integral can then be rewritten

$$2 \int \frac{du (u+\frac{r}{\beta})}{(1+(u+\frac{r}{\beta})^2)^2} \sqrt{-\gamma-u^2}$$

Mathematica returns a huge mess to this. I think the fact that the u^2 is negative inside the radical makes this a different form from what you might have been thinking. I am not sure this form is easier to evaluate...

I think I may be able to construct a different integrand if I parameterize the problem differently...will check back later.