$\int_0^{2\pi}-e^{\cos(t)}[\sin(t)\cdot \cos(\sin(t))+\cos(t)\cdot \sin(\sin(t)]\,dt$
2026-03-26 22:17:20.1774563440
I would like to compute the integral
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Notice that the integral takes the form of $$\int_0^{2\pi} \frac{d}{dx} [f(x)\cdot g(x)] \; dx$$ where $f(x)=e^{\cos{(t)}}$ and $g(x)=\cos{\left(\sin{t}\right)}$.
$$\int_0^{2\pi}-e^{\cos(t)}[\sin(t)\cdot \cos(\sin(t))+\cos(t)\cdot \sin(\sin(t)]\;dt=e^{\cos{(t)}}\cdot \cos{\left(\sin{t}\right)} \big \rvert_0^{2\pi}=\boxed{0}$$