Primitive of Arcsine of x over a over x squared

Theorem

 * $\ds \int \frac {\arcsin \frac x a \rd x} {x^2} = \frac {-\arcsin \frac x a} x - \frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$

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

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\arccos \dfrac x a} {x^2}$


 * Primitive of $\dfrac {\arctan \dfrac x a} {x^2}$


 * Primitive of $\dfrac {\arccot \dfrac x a} {x^2}$


 * Primitive of $\dfrac {\arcsec \dfrac x a} {x^2}$


 * Primitive of $\dfrac {\arccsc \dfrac x a} {x^2}$