∫ sin(2t)*e^[cos(t)] dt (from t=0 to π)

My work:

trig identity: sin2t= 2sintcost

bring constant to the outside

=> 2 ∫ sin(t)cos(t)*e^[cos(t)] dt (from t=0 to π)

I did u substitution and did

u=cos(t)

du=-sin(t)dt

new intervals:

cos(pi)=-1

cos(0)=1

Im not sure what to after this. Do I divide du=-sint(dt) by negative 1 and bring that -1 to the outside of integral since there is only a sin(t)dt not -sin(t)dt?

I appreciate any help!