Calculating E-field of Uniformly Charged Sphere

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I need to calculate the electric field both inside and outside a sphere with a charge density p. The trick is I am required to use Coulomb's Law. I cannot appeal to Gauss' Law.

My instructor suggests considering the electric field due to a ring of charge. Then to use that result to find the electric field of a spherical shell. Finally, I will use that result to find the electric field due to a sphere.

I've done the first part with no problems, and my answer checks with the book. (R=radius of ring, r=distance along axis of ring)
[tex]E=\frac{Qr}{4 \pi \epsilon_0 (r^2+R^2)^{3/2}}[/tex]
However, I am stuck on how to use this result to find the E field due to a spherical shell.

I begin by considering a "ring element" of radius r and I consider a point on the x-axis (which happens to coincide with the axis of the ring). I need to find the dE associated with this ring element, but it's not so simple. It should be [tex]dE=\frac{dQ(d-x)}{4\pi\epsilon_0((d-x)^2+r^2)^{3/2}}[/tex] (d=x-coordinate of given point on x-axis, x=x-coordinate of plane through ring, r=ring radius) (this isn't showing up correctly: it's missing "dE" and "dQ") Here dQ, x, r are all changing and when I plug in the relations between them I get an ugly integral, that doesn't appear to come out with the 1/r^2 dependence. Any suggestions?
 
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