Integral of ln2x?
I will denote an integral as \int (LaTeX).
We can let u = 2x and du = 2dx, and substitute
\int ln (2x) dx = (1/2) \int ln u du. Either using integration
by parts or by memorization, this is equal to
(1/2) u ln u - u + C = (1/2)(2x ln (2x) - 2x) + C, where C is an
arbitrary constant.