Primitive of x squared by Root of a squared minus x squared

Theorem

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

Proof

 $\displaystyle z$ $=$ $\displaystyle x^2$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d z} {\d x}$ $=$ $\displaystyle 2 x$ Power Rule for Derivatives $\displaystyle \leadsto \ \$ $\displaystyle \int x^2 \sqrt {a^2 - x^2} \rd x$ $=$ $\displaystyle \int \frac {z \sqrt {a^2 - z} \rd z} {2 \sqrt z}$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \sqrt z \sqrt {a^2 - z} \rd z$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {\frac {-2 z + a^2} 4 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \int \frac {\d z} {\sqrt z \sqrt {a^2 - z} } } + C$ Primitive of $\sqrt {\paren {a x + b} \paren {p x + q} }$ $\displaystyle$ $=$ $\displaystyle \frac {-2 z + a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} {16} \paren {2 \sinh^{-1} \sqrt {\frac z {a^2} } } + C$ Primitive of $\dfrac 1 {\sqrt {\paren {a x + b} \paren {p x + q} } }$ $\displaystyle$ $=$ $\displaystyle \frac {-2 z + 2 a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \sinh^{-1} \sqrt {\frac z {a^2} } + C$ $\displaystyle$ $=$ $\displaystyle \frac {a^2 - z} 4 \sqrt z \sqrt {a^2 - z} + \frac {a^2} 8 \sqrt z \sqrt {a^2 - z} + \frac {a^4} 8 \sinh^{-1} \frac {\sqrt z} a + C$ $\displaystyle$ $=$ $\displaystyle \frac {-x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C$ substituting for $z$

$\blacksquare$

Sources

(in which a mistake apppears)