Answer
E = $\frac{1}{4πε_{0}}$$\frac{λ(2πr)z}{(r^{2}+z^{2})^{3/2}}$ ẑ
Work Step by Step
Horizontal components cancel, leaving E = $\frac{1}{4πε_{0}}\times ∫[\frac{λdl}{x^{2}}cos\theta]$ ẑ Where x is script r, or the separation vector.
$x^{2} = r^{2}+z^{2}$, $cos\theta = \frac{z}{x}$ (both constants), while $∫dl = 2πr$.
