# 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 – Omnomnomnom Feb 21 '15 at 19:32
• But the function is not entire, so how can I use Cauchy's Integral Formula? – nonremovable Feb 21 '15 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. – Omnomnomnom Feb 21 '15 at 19:42
• Ah! Thank you for that last comment. I get it now. – nonremovable Feb 21 '15 at 19:45

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