This is a problem in electromagnetics applying coulomb's law, in brief, we have a disc(wire) with charges on it. We want to calculate the electric field at a point on the axis of the disc. The book had a rather annoying way of solving the problem, so I wanted to go with something simpler and went with the following:

. R is a unit vector

.

.

when we put all of that together, we get:

.

But unless we take away the 2 s from the vector we won't get to the right answer which is .

Note: integral limits and .

What did I do wrong?