Integrate [x^3/sqrt(1-x^2/k^2)]dx

Can someone please check my process, also please can you advise on an easier way to write mathematical notation on this forum?

Integrate ( x^3/(1-x^2/k^2))dx

where k is a constant.

Let u = x^2, dv = x/sqrt(1-x^2/k^2)

Integration by parts.

Integral fx = uv - integral vdu

v = -k^2*sqrt(1-x^2/k^2)

du = 2xdx

susbstituting for u,v,du

Integral fx = -k^2 *x^2*sqrt(1-x^2/k^2)+ integral [2xk^2*sqrt(1-x^2/k^2)dx]

Integral fx = -k^2 *x^2*sqrt(1-x^2/k^2) - 2/3*k^4*((1-x^2/k^2)^3/2)