Calculate $\int_\gamma\frac{|z|e^z}{z^2}dz$, where $\gamma$ is the circle of radius 2 centered at the origin. Calculate $$\int_\gamma\frac{|z|e^z}{z^2}dz,$$ where $\gamma$ is the circle of radius 2 centered at $0\in\mathbb{C}$.
I wanted to apply the Residue Theorem, but I think it is not possible because $|z|$ is not analytic at the origin. Also, I tried to calculate the integral using the parametrization $z=2e^{i\theta}$, $0\leq\theta\leq 2\pi$. Then $$\int_\gamma\frac{|z|e^z}{z^2}dz=\int_0^{2\pi}\frac{|2e^{i\theta}|e^{2e^{i\theta}}}{2e^{2i\theta}}2ie^{i\theta}d\theta= 2i\int_0^{2\pi}\frac{e^{2e^{i\theta}}}{e^{i\theta}}d\theta.$$ But I have not been able to solve that integral.
I thank any suggestion.

As @leoli1 and @Gary hint,$$\int_{|z=2|}\frac{|z|e^zdz}{z^2}=\int_{|z=2|}\frac{2e^zdz}{z^2}=2\pi i\operatorname{Res}_{z=0}\frac{2e^z}{z^2}=4\pi i.$$Alternatively,$$2i\int_0^{2\pi}\frac{e^{2e^{i\theta}}}{e^{i\theta}}d\theta=2i\int_0^{2\pi}2d\theta=4\pi i$$because $\int_0^{2\pi}e^{ik\theta}d\theta=2\pi\delta_{k0}$ for $k\in\Bbb Z$.
