Like Melody, I'm on shaky ground here....so.....you might have to take this with a grain of salt....
By the Integral Test, the function in the integrand must be eventually positive and decreasing on the interval of interest - in this case, 0 to ∞.
Here's the graph of y = cos(x)*e^(-sin(x))
.....https://www.desmos.com/calculator/ho13rj5sjr
Note that this function is not eventually positive and decreasing on the interval (it is periodic).....thus.....the integral diverges
That's my two cents worth.....
Alan??? heureka???....to the rescue???
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