Primitive of x squared by Arcsine of x over a

Theorem

 * $\displaystyle \int x^2 \arcsin \frac x a \rd x = \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$

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

let:

and let:

Then:

Also see

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


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


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


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


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