# Primitive of x squared by Arcsine of x over a

## Theorem

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

## Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

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

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds x^2$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds \frac {x^3} 3$ Primitive of Power

Then:

 $\ds \int x^2 \arcsin \frac x a \rd x$ $=$ $\ds \frac {x^3} 3 \arcsin \frac x a - \int \frac {x^3} 3 \paren {\frac 1 {\sqrt {a^2 - x^2} } } \rd x + C$ Integration by Parts $\ds$ $=$ $\ds \frac {x^3} 3 \arcsin \frac x a - \frac 1 3 \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } + C$ Primitive of Constant Multiple of Function $\ds$ $=$ $\ds \frac {x^3} 3 \arcsin \frac x a - \frac 1 3 \paren {\frac {\paren {\sqrt {a^2 - x^2} }^3} 3 - a^2 \sqrt {a^2 - x^2} } + C$ Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$ $\ds$ $=$ $\ds \frac {x^3} 3 \arcsin \frac x a + \frac {\paren {x^2 + 2 a^2} \sqrt {a^2 - x^2} } 9 + C$ simplifying

$\blacksquare$