I would use "integration by parts" to solve this one.
The rule goes as follows:
int f(x) g'(x) "d"x = f(x) g(x) - int f'(x) g(x) "d"x
So, out of the product x * ln(2x) you need to pick one of the factors to differentiate and one to integrate. If ln is one of your factors, it is usually a good bet to choose this factor to be differentiated.
So, I would pick f(x) = ln(2x) and g'(x) = x.
Now, differentiate f(x) and integrate g(x) to compute f'(x) and g(x):
f(x) = ln(2x) => f'(x) = 1/(2x) * 2 = 1/x
g'(x) = x => g(x) = 1/2 x^2
Now, apply the rule:
int x ln(2x) "d"x
= ln(2x) * 1/2 x^2 - int 1/2 x^2 * 1 / x "d"x
= 1/2 x^2 ln(2x) - int 1/2 x "d"x
= 1/2 x^2 ln(2x) - 1/2 (1/2x^2)
= 1/2 x^2 ln(2x) - 1/4 x^2
I hope that this helped!
