Ideas as to how I should start to integrate this one?

Yeah, just factor out x^2, so you have
[tex]\int x\sqrt{C^2-x^2}dx[/tex]
Then just make the substitution [itex]u=x^2[/itex].

 

You can make another substitution, e.g. [tex]z = C^2 - u [/tex] or just make your original substitution [tex]u = C^2 - x^2 [/tex], which gives

[tex]du = -2x dx[/tex]

[tex]\frac{du}{-2x} = dx [/tex]

then you just use your normal integration rules.

 

Just a minor correction.
We should have:
[tex]\int\sqrt{C^{2}x^{2}-x^{4}}dx=\int|x|\sqrt{C^{2}-x^{2}}dx[/tex]