$\int_{0}^{\pi} \exp\left(\cos\left(t\right)\right)\cos\left(\sin\left(t\right)\right)\,{\rm d}t=\pi$ Does anyone have a proof of the above integral? I have one proof, but I wanted to see other proofs.

Hint:
By some simple change of variables we get
$$\int_0^\pi e^{\cos t}\cos\sin tdt=\frac{1}{2}\int_0^{2\pi} e^{\cos \theta}\cos \sin\theta d\theta=\mathrm{Re}\left(\frac{1}{2}\int_0^{2\pi}e^{e^{i\theta}}d\theta\right) = \pi\cdot\mathrm{Re}\left(\frac{1}{2\pi i}\int_{C}\frac{e^z}{z}dz\right),$$
where $C$ is the unit circle. Now use Cauchy integral formula.
