Primitive of Arcsine of x over a over x squared

Theorem

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

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 $\dfrac 1 {x^2} \arccos \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \arctan \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \arccot \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \arcsec \dfrac x a$


 * Primitive of $\dfrac 1 {x^2} \arccsc \dfrac x a$