Integration by part.
{2xcos2xdx
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Looks like the bot won't even attempt this one
btw
integral 2xcos2xdx = xsin(2x) + (1/2)cos(2x) + c
to show the steps of the traditional method would be hard to type out
in this format.
I suggest you look at some excellent videos by "blackpen redpen "
who has a very nice algorithm for integration by parts
Google "integration by parts, DI method, VERY EASY" ,you will like it
He does one similar to yours
∫ 2x cos2x dx
Let u=x, du = dx
dv = 2cos2x dx, v = sin2x
∫ 2x cos2x dx = x sin2x - ∫ sin2x dx
= x sin2x + 1/2 cos2x + C