Primitive of Arcsine of x over a/Proof 2

Theorem

 * $\displaystyle \int \arcsin \frac x a \ \mathrm d x = x \arcsin \frac x a + \sqrt {a^2 - x^2} + C$

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then: