I've currently trying to learn electromagnetism in depth and I tried to solve the **Biot-Savart** law, for a magnetic field generated by a current.
$$
\vec{B} = \frac{\mu_0}{4 \pi} \int{\frac{I \; \hat{r} \times \mathrm{d} \vec{\ell}}{r^2}}
$$
When I looked up for information on how to solve the equation, there are always simplifications, like:
$$
B = \frac{\mu_0}{4 \pi} \int{\frac{I \; \mathrm{d} \ell \sin{\theta}}{r^2}} \rightarrow B = \frac{\mu_0 I}{2 \pi R}
$$


**Edit:**
But it is possible to solve this integral in a vectorial form and without the need of simplification for the magnetic field in a straight wire carrying a current? Would you be able to arrive to a solution similar to the one of the electric field generated by an electric current in a straight wire $\frac{\lambda}{2 \pi \varepsilon_0 r} \hat{r}$ (which you arrive using Gauss's law)?