Answer
$$\int e^{\cos t}\sin tdt=-e^{\cos t}+c$$
Work Step by Step
To evaluate the integral $$\int e^{\cos t}\sin tdt$$
we will use substitution $\cos t=z$ which gives us $-\sin tdt=dz\Rightarrow \sin tdt=-dz$. Putting this into the integral we get:
$$\int e^{\cos t}\sin tdt=\int e^z(-dz)=-e^z+c$$
where $c$ is arbitrary constant. Now we have to express solution in terms of $t$:
$$\int e^{\cos t}\sin tdt=-e^z+c=-e^{\cos t}+c$$