# How to integrate this

I'm trying to integrate $$F_0a^2(-\frac{a}{2})\frac{1}{(\rho^2+\frac{a^2}{4})^{3/2}}$$ This is in cylindrical coordinates, so $\rho$ represents a radius and I want to integrate the expression over a disk with radius a. The correct answer is $$-2\pi F_0a^2(1-\frac{1}{\sqrt{5}})$$ How can I find this answer? Any pointers would be much appreciated. I don't think sharing my attempts would do any good, I don't have much faith in them. But basically I've tried using a table of indefinite integral because I don't know how to start from scratch on this.

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Make a substitution $u=\rho^2+\tfrac{a^2}{4}$. Don't forget the polar Jacobian $dA=\rho\,d\rho d\varphi$.
With that substition, and the Jacobian, I'll have to integrate $\frac{ \sqrt{u-\frac{a^2}{4}}} {u^{3/2}}$ which is not easy either, I think. –  Pickett Jan 6 '13 at 1:06
the integrand becomes $du/(2u^{3/2})$ –  Jonathan Jan 6 '13 at 1:07