I need to solve the magnetic field created by a wire with an arc shape. This integral is very complicated, so I decided to approximate the arc into circular chords and calculate the magnetic field of these small segments. The integral to solve looks like $$\int \frac{dy'}{[z^2+(x-x')^2+(y-y')^2]^{3/2}}$$ and considering that the segments have a general orientation $$ y'=px'+b $$
This integral is difficult for Mathematica, so it gives me a conditional expression with a lot of assumptions, which I believe is wrong because it doesn't match my numerical result. My new idea is to calculate the magnetic field of a vertical segment perpendicular to the, let's say, x axis, so the integral would be easier to solve:
$$\int \frac{dy'}{[z^2+(x-x_0)^2+(y-y')^2]^{3/2}}$$
I want to use the result of this integral to, by translations and rotations, calculate the magnetic field of the different circular chords. How could you make this? Thanks.
Hint: $\oint_S B\cdot dr = \mu_0 I_{encl}$ and you can parameterize the integral by taking a circular path of arbitrary radius.