∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^z]  
		=   sin(Aθpc(POIo(t),t=0))^(z+1)/(z+1)  

The following list makes it easier to copy/paste...  Note that the result = 0 for the lower limit of integration, leaving only the upper end.
/%
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))]  								= sin(Aθpc(POIo(t),t=0))^1/1 
(endMath)

	∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))] 
	= -	{cos(Aθpc(POIo(t),t=0)) - cos(0)}
	=(-1)*{cos(Aθpc(POIo(t),t=0)) - 1}

(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))]  								= {1 - cos(Aθpc(POIo(t),t=0)}
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1]  = sin(Aθpc(POIo(t),t=0))^2/2 
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^2]  = sin(Aθpc(POIo(t),t=0))^3/3
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3]  = sin(Aθpc(POIo(t),t=0))^4/4
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^4]  = sin(Aθpc(POIo(t),t=0))^5/5
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5]  = sin(Aθpc(POIo(t),t=0))^6/6
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^6]  = sin(Aθpc(POIo(t),t=0))^7/7
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^7]  = sin(Aθpc(POIo(t),t=0))^8/8
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^8]  = sin(Aθpc(POIo(t),t=0))^9/9
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^9]  = sin(Aθpc(POIo(t),t=0))^10/10
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^10] = sin(Aθpc(POIo(t),t=0))^11/11
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^11] = sin(Aθpc(POIo(t),t=0))^12/12
(endMath)