# complex integral: $\int_C \frac{e^z+\sin{z}}{z}dz$

Evaluate: $$\int_C \frac{e^z+\sin{z}}{z}dz$$ where, $$C$$ is the circle $$|z|=5$$ traversed once in the counterclockwise direction.

I can't find an anti-derivative to this function, and I am not sure one exists. I was thinking about using several theorems but each come up short. For instance, I cannot use the Closed Curve Theorem since the function in the integrand is not entire (at least I don't think it is).

Any hints (to help get me started)?

• For example, you could use Cauchy's integral formula Feb 21, 2015 at 19:32
• But the function is not entire, so how can I use Cauchy's Integral Formula? Feb 21, 2015 at 19:40
• The numerator is entire, and that's all you need for the formula. Don't confuse this with Cauchy's integral theorem, which is specifically about the integral of entire functions. Feb 21, 2015 at 19:42
• Ah! Thank you for that last comment. I get it now. Feb 21, 2015 at 19:45

Let $f(z) = e^z + \sin z$, then by Cauchy's Integral Formula we have
$$2\pi i f(0) = \int_C \frac{e^z + \sin z }{z - 0} dz \implies \int_C \frac{e^z + \sin z }{z - 0} dz = \color{#f03}{ 2\pi i}$$