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I'm struggling to solve such an integral using only the definition of the integral of complex function, any hints?

$ \int_{\gamma} \frac{dz}{z^2+4iz}. $

contour $\gamma$ is a triangle with vertices -1+2i, -1-2i and 1

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By Cauchy's Residue theorem: since the function $\,\displaystyle{f(z)=\frac{1}{z^2+4iz}}\,$ only has the pole $\,z=0\,$ inside the domain enclosed by the triangle's perimeter and the function's analytic over the perimeter, we get

$Res_{z=0}(f)=\lim_{z\to 0}zf(z)=\frac{1}{4i}\Longrightarrow\int_\gamma\frac{dz}{z^2+4iz}=2\pi i\frac{1}{4i}=\frac{\pi}{2}$

If by "definition" you meant the line integrals on the different sides of the triangles then that's way too cumbersome and lengthy (at least for me)