Use "integration by parts".
The formula is
#int f(x) g'(x) "d" x = f(x) g(x) - int f'(x) g(x) "d"x#
So, in your product #x * sin(6x)# you need to determine which factor you would like to integrate and which factor you would like to differentiate.
As you have a #sin# function as one of the arguments, I would argue that it is usually a good bet to pick it as the one to integrate since integrating it doesn't make it more complicated.
So, let's say #f(x) = x# and #g'(x) = sin(6x)#.
As next, let's differentiate #f(x)# and integrate #g'(x)# in order to compute #f'(x)# and #g(x)#, respectively.
#f(x) = x => f'(x) = 1#
#g'(x) = sin(6x) => g(x) = - 1/6 cos(6x)#
Now, it's time to apply the formula:
#int x sin(6x) "d"x#
#= x * (- 1/6 cos(6x)) - int 1 * (-1/6) cos(6x)) "d"x#
# = - 1/6 x cos(6x) + 1/6 int cos(6x) "d"x#
# = - 1/6 x cos(6x) + 1/6 * 1/6 * sin(6x)#
# = - 1/6 x cos(6x) + 1/36 * sin(6x)#
