What is the geometric significance of integration in the complex plane?

In BC Calc we learned something called "complex integration" where if we were integrating something like $ \displaystyle \int e^x \cos x \, dx $, we could rewrite as $ \mathfrak{Re} \left( \displaystyle \int e^x e^{ix} \, dx \right) $. What does it mean geometrically when we integrate with complex numbers and then take the real part?

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$