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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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up vote 2 down vote accepted

Make a substitution $u=\rho^2+\tfrac{a^2}{4}$. Don't forget the polar Jacobian $dA=\rho\,d\rho d\varphi$.

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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. – user3680 Jan 6 '13 at 1:06
    
the integrand becomes $du/(2u^{3/2})$ – Jonathan Jan 6 '13 at 1:07
    
Yes you are right of course. Solved it now, thanks a lot. – user3680 Jan 6 '13 at 1:21

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