Double integral of R (x(sin(x+y), R= [0,pi/6] X [0,pi/3]
Thanks in advance.
$\displaystyle \begin{aligned}\int_0^{\pi/6}\int_0^{\pi/3}x\sin(x+y)\,dydx &= \int_0^{\pi/6}\Bigl[-x\cos(x+y)\Bigr]_0^{\pi/3}dx \\ &= \int_0^{\pi/6}\bigl(x\cos x-x\cos(x+\tfrac\pi3)\bigr)dx.\end{aligned}$
Now you have a single integral that you can integrate by parts.