$\sin(e^x) +c $
Work Step by Step
Recall that $(e^x)'=e^x$ Recall that $(\sin x)'=\cos x$. Let $ u=e^x $, then $ du=e^x dx $ and hence we have $$ \int e^x \cos (e^x) dx=\int \cos (u) du= \sin(u) +c=\sin(e^x) +c . $$
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