Let x and a represent distance. Is the equation ∫ dx a^2 - x^2 =...

The dimension of $dx$ is $[L]$.

The dimension of $\sqrt{a^2 - x^2}$ is $\sqrt{[L^2]} = [L]$.

The dimension of the left-hand side is $\dfrac{[L]}{[L]} = [M^0 L^0 T^0]$, which is dimensionless.

On the right-hand side, the argument of the inverse sine function, $\dfrac{a}{x}$, is dimensionless. The inverse sine function itself represents an angle, which is also dimensionless.

The term $\dfrac{1}{a}$ has the dimension $[L^{-1}]$.

Therefore, the dimension of the right-hand side is $[L^{-1}]$.

Since the left-hand side is dimensionless and the right-hand side has dimensions of $[L^{-1}]$, the equation is not dimensionally correct.