For the next integral I obtained and as my bounds which is incorrect according to the solution I was given, can someone lead me from here?
Did you draw the region?
See the pic. You have regions D1, D2 and D3. In D1 and D3 you integrate cos(x + y) while in region D2 (for which you will need two integrals) you would integrate -cos(x + y)
the red line is the line and the blue line is the line