Integrate (sinx- cosx- sec^(2)x + csc^(2)*(4x))dx show work?

This is a just very basic piece by piece integration. You should have a sheet in the front of your textbook with common trig integrals and derivatives. The answer is cosx + sinx - tanx - 1/4 cot (4x). The first three are just direct application of term by term integration the last one you use the u sub of 4x=u which is why it's 1/4 cot. (because du=4dx, so du/4=dx)