Primitive of Power of x by Arcsine of x over a

Theorem

 * $\displaystyle \int x^m \arcsin \frac x a \ \mathrm d x = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \ \mathrm d x} {\sqrt{a^2 - x^2} } + C$

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:

Also see

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


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


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


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


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