Complex Analysis: Another Integral Snack

Today, we use complex analysis to evaluate another difficult-looking integral.
Cauchy's Integral Theorem:

[violintegral]

Believe or not, this function actually has a simple elementary antiderivative. Replacing theta with x it is -exp(cos(x))*cos(sin(x)) + C. This becomes clearer when sin(x + sin(x)) is expanded using the sine angle sum formula, since it gives a function that looks like a derivative by the product rule.

[wuyqrbt]

Great ... Of course I like it.
Nice example, with unexpected result (it's possible we turn the answer in terms of π ? Its answer is so familiar for me! )
Thank you so much dear *QN^3* .
Please more example like this.

[captainchicky]

this is rather similar to the integral of e^cosx * cos(sinx) from 0 to 2pi i found a while back