Answer
$\cos e^{-x}+C$
Work Step by Step
Our aim is to solve the integral $ \int e^{-x} \sin (e^{-x}) \ dx$
Let us consider that $a =e^{-x}$ and $\dfrac{da}{dx}=- e^{-x} \implies dx=\dfrac{-1}{e^{-x}} \ da$
Now, $ \int e^{-x} \sin (e^{-x}) \ dx = \int e^{-x} \sin a (\dfrac{-1}{ e^{-x}} ) \ da$
or, $=- \int \sin a \ da $
or, $=-(-\cos a)+C$
or, $=\cos e^{-x}+C$
