# Primitive of Power of x by Arcsine of x over a

## Theorem

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

## 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:

 $\displaystyle u$ $=$ $\displaystyle \arcsin \frac x a$ $\displaystyle \implies \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle \frac 1 {\sqrt {a^2 - x^2} }$ Derivative of $\arcsin \dfrac x a$

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle x^m$ $\displaystyle \implies \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {x^{m + 1} } {m + 1}$ Primitive of Power

Then:

 $\displaystyle \int x^m \arcsin \frac x a \rd x$ $=$ $\displaystyle \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \int \frac {x^{m + 1} } {m + 1} \left({\frac 1 {\sqrt {a^2 - x^2} } }\right) \rd x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt{a^2 - x^2} } + C$ Primitive of Constant Multiple of Function

$\blacksquare$