cleared the fraction in the radical to make life easier ...
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)
Rewrite the integral as following:
Let and (using the chain rule) differentiate this with respect to :
Solving this for , as our purpose was, we have:
Going back to our original integral and putting that in for we get:
Finding in terms of from the relation of our substitution gives us:
.
Putting that in for , we get a simple function in the integrand that is easy to integrate:
Substituting back for what we have let to be, we finally get:
.