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)
