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Study Resources (Calculus)

Ordinary Differential Equations problem. Please show work and thanks!! Assume that the heating and cooling of the Goessmann building at the University of Massachusetts, Amherst, can be modeled by the following ODE: Dt/dt=k_1(T-T_a) + k_2(T-T_omega)+P, where T = T(t) is the temperature in the building at time.
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On my last mid-term I had to integrate csc(t)dt. It was multiple choice and the "correct answer" was ln|csc(t)-cot(t)|+C.  Everywhere I searched online says this is equal to -ln|csc(t)-cot(t)|+C. Also I had to ingtegrate cos(7t)sin(2t).  The correct answer was 1/2(sin(9t)-sin(5t)).  Online I saw sinxcosy=1/2(sin(x+y)+sin(x-y)). Please explain which.
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