How to use dimensional analysis in an integral

How would you find the answer using dimensional analysis and what is it?

Check out the denominator of the integrated function. Since we have $2ax-x^2$ there and since one cannot add apples and oranges, we can immediately see that the unit of $a$ is the same as the unit of $x$. Let us say it is meters just for the sake of definiteness. The function you are integrating is $f(x)=\frac{1}{\sqrt{2ax-x^2}}$, which is meters$^{-1}$, and since integrating wrt $x$ is akin to finding the area below $f(x)$, the integral should then be meters$^{-1}\times$meters, i.e. non-dimensional.
The argument of $\sin^{-1}$ is clearly non-dimensinal, so is the value of $\sin^{-1}(x/a-1)$. Thus, the only way for the answer to be non-dimensional is if $n=0$.