Integrate $\int_{0}^{2\pi}e^{\cos\theta}\cos(\sin\theta)d\theta$

Question

Please I need assistance in solving the integral below:

$$\int_{0}^{2\pi}e^{\cos\theta}\cos(\sin\theta)d\theta$$

I tried solving by making these substitutions: $$u=\sin\theta$$ OR $$u=\cos\theta$$ But i ended up in a deadlock.

Answer 1

Your integral can be written as real part of $\int_0^{2\pi} d\theta \exp{(z)}$, with $z=\exp{(I \theta)}$. Now, change variables to $dz = I z d\theta$, and your integral is the real part of $-I \oint \frac{\exp{( z)}}{z}$ (with a contour the circle or radius $1$ around the origin) that is equal to $2\pi$ from the theorem of residues (Cauchy). Hence, your integral is $2\pi$.