Primitive of x cubed over Root of x squared minus a squared

Theorem

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

for $\size x > a$.

Proof
With a view to expressing the problem 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:

and let:

Then:

Also see

 * Primitive of $\dfrac {x^3} {\sqrt {x^2 + a^2} }$
 * Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$