Need a guideline for proving/deriving magnetic field from current on a sheet

[BlackMelon]

Homework Statement

From the attached picture, I want to proof that Hx=0.5Ky when z>0 and Hx = -0.5Ky when z<0 (Hx is magnetic field caused by surface current density K)
I get stuck with integral problem so my questions are...
1) Can I proof by this Biot-Savart Law?
2) If so, how to eliminate the unit vector a_ρ or convert it into x or y

Homework Equations

The first equation in the attached picture

The Attempt at a Solution

Look at the attached picture, I was trying to solve but got stuck on math

PS: The problem is at the bottom of the page 192 and 193 of this book:
https://www.mediafire.com/?f0cpqvavy1ddl29
PS2: In the picture, rho should be sqrt(x^2+z^2) not sqrt(x^2+y^2), sorry for the mistake

 

[Guneykan Ozgul]

You can use Ampere's law for an easy solution.

 

[berkeman]

You can use Ampere's law for an easy solution.

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