I'm having a test on integration on Monday and I'm having some major problems with solving integrations involving trigonometric substitution. There's a sample of a problem I'm having issues with:

Code:

*I* = Integration symbol
u = dummy variable for substitution
*I* tan^2(6x) * dx
My work:
sin^2(6x)
*I* ---------- * dx u = cos(6x)
cos^2(6x) du = -6sin(6x) * dx
du
dx = ------------
-6sin(6x)
sin^2(6x) du
*I* ---------- * --------
u -6sin(6x)
-1 sin(6x)
--- *I* -------- * du
6 u

This is where I get stuck. I remember my teacher saying that I have to get everything in terms of u before I go substituting u back into the equation, however, I don't see a feasible way to do this. If someone could step my through this one so I can see how it's done, I'd really appreciate it.

If you all need any more information about the problem or something I'll do my best to provide. Regards.