How to solve the Biot-Savart Law?

Question

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: 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)?

Answer 1

What about:

$$d \vec{B} = \frac{\mu_0 I d \vec{\ell} \times \vec{r}}{4 \pi r^3}$$

or

$$d \vec{B} = \frac{\mu_0 I d \vec{\ell} \times \hat{r}}{4 \pi r^2}$$

$B$ = magnetic flux density; $I$ = current in wire; $\ell$ = length of wire; $r$ = distance of a point in the field to a segment of the wire; $\mu_0$ = magnetic permeability of free space.