Cool Integration Techniques! (Part 1)

Let $F (x) = {e}^{cosx} . cos(sinx) \\ \qquad = R.P.O \quad {e}^{cosx} . {e}^{isinx} \\ \qquad = R.P.O \quad {e}^{cosx + isinx} \\ \qquad = R.P.O \quad {e}^{{e}^{ix}}\\ \qquad = R.P.O \quad 1 + {e}^{ix} + \frac{ {e}^{2ix}}{2!} + \frac{{e}^{3ix}}{3!} + .......\\$

$R.P.O ==$ Real Part of

Hence you will get

$\int\limits_{0}^{2\pi}{F (x)} = \int\limits_{0}^{2\pi}{1 + cosx + \frac{cos2x}{2!} + \frac{cos3x}{3!} + .....} = 2\pi + 0 + 0 + .... = 2\pi$

The real part of the integral integral can be written as $\displaystyle I=\int_{0}^{2\pi}e^{e^{i\theta}} d\theta$

Now putting $e^{i\theta}=z$ and integrating around the unit circle we get,

$\displaystyle I=\int_{|z|=1} \frac{-ie^z}{z}dz=2\pi i\times(-i)=\boxed{2\pi}$

Alternate solution could be resorting to Feynman's way of Integration by considering the function $F(x)=\large \int_{0}^{2\pi}{{e}^{x\cos\theta} \cos(x\sin\theta) \, d\theta}$ and then proceeding.

Our required integrand is $F(1)$ .