How to use dimensional analysis in an integral

Question

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

The value of n is

(a) 0

(b) -1

(c) 1

(d) none of these.

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

Answer 1

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 $n=0$.