$\int \frac{\sqrt{a^2-x^2}}{x^2}dx$ using trigonometric substitition

Question

I'm aware this question is more easily done using substitution by parts or euler substitution, but this was under a section in my book where we were asked to use trigonemtric substations. Substituting $$x=a\sin{u}$$ I eventually get to $$-\cot{u}-u$$ But I'm not sure how to express this in terms of $x$.

Answer 1

$\int \frac{\sqrt{a^2-x^2}}{x^2}dx$

$x=a \sin u. dx = a\cos u du.$

$\int \frac{a \cos u}{a^2\sin^2 u} a \cos u du=\int \cot^2 u \ du=\int \csc^2 u -1 \ du= -\cot u-u+C$

$x/a= \sin u. $

$\sqrt{1-x^2/a^2}= \cos u$

$\frac{-\sqrt{a^2-x^2}}{x}-\arcsin (x/a)+ C$

Since these are trig functions, there are other possible, equivalent solutions.