A unitless quantity

$\int \frac{dx}{\sqrt{2ax-x^2}}=a^{\mathrm{n}}{\sin }^{-1}\left[\frac{x}{a}-1\right]$

Find the value of $n$. (Solve using dimensional analysis)

Find the dimensions of

(a) angular speed $\omega$

(b) angular acceleration $\alpha$
(c) torque $\Gamma$ and
(d) moment of inertia $I$.
Some of the equations involving these quantities are

$\omega =\frac{{\theta }_2-{\theta }_1}{t_2-t_1}, \alpha =\frac{{\omega }_2-{\omega }_1}{t_2-t_1}, \Gamma =\overrightarrow{r}\times \overrightarrow{F}\text{ and }I=m{r}^2$.
The symbols have standard meanings.

Find the dimensions of
(a) electric field $E$,

(b) magnetic field $B$ and

(c) magnetic permeability ${\mu }_0$.
The relevant equations are $F=qE, F=qvB,$ and $B=\frac{{\mu }_{0}I}{2\pi a}$, where $F$ is force, $q$ is charge, $v$ is speed, $I$ is current, and $a$ is distance.

Find the dimensions of

(a) electric dipole moment, $p$ and

(b) magnetic dipole moment, $M$.

The defining equations are $p=q·d$ and $M=I A$, where $d$ is distance, $A$ is area, $q$ is charge and $I$ is current.