Your mistake is here:
u = sinx du = dx
It should be this way:
u = sin(x) => du = cos(x) dx holds.
But you should do more that above substitution. Use some trigonmetric relations to introduce a cos(x) in the integral (for example sin(2x)=2sin(x)cos(x))
I am just feeling that then you can use integration by parts to get the answer, which is:
1/2 (x - sin(x)cos(x))