Sin(2x)=2sin(x)cos(x).
Let u=sin(x); du=cos(x)dx.
S[0,(pi)/2](e^sin(x)sin2xdx) denotes the integral in the species range.
This can now be written: S[0,1](e^u.2udu)=2S[0,1](ue^udu).
Let p=u, dp=du; dq=e^udu, q=e^u. d(pq)=qdp + pdq; d(ue^u)=e^udu+ue^udu.
Integrating: ue^u=e^u+S(ue^udu); 2S[0,1](ue^udu)=2e^u(u-1)[0,1]=2(0-(-1))=2.