# Primitive of x squared by Root of a squared minus x squared/Mistake

## Source Work

Chapter $14$: Indefinite Integrals
Integrals involving $\sqrt {a^2 - x^2}$: $14.246$

This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).

## Mistake

$\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$

As demonstrated in Primitive of $x^2 \sqrt {a^2 - x^2}$ the correct expression is in fact:

$\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C$