Cauchy's Integral Theorem (also known as Cauchy's Fundamental Theorem) is a cornerstone in complex analysis. It states that if $f(z)$ is a holomorphic (analytic) function on and inside a simple closed curve $C$, then the contour integral of $f(z)$ around $C$ is zero:

$\int_C f(z) \, dz = 0$

Conditions for Cauchy's Theorem:

1. $f(z)$ must be holomorphic (analytic) inside and on the curve $C$.
2. $C$ is a positively oriented, simple closed contour (meaning it encloses a region, typically traversed counterclockwise).

Justification for the Function $f(z) = z^3 + i z^2 - 5z + 2i$:

Given the function $f(z) = z^3 + iz^2 - 5z + 2i$ and the contour $C$ as the circle $|z| = 1$, we need to verify whether $f(z)$ satisfies the conditions of Cauchy's Theorem:

1. Analyticity:
   $f(z)$ is a polynomial in $z$, which is holomorphic (analytic) everywhere in the complex plane. Specifically, it is analytic inside and on the contour $C$ (the circle $|z| = 1$). There are no singularities inside or on $C$.

2. Closed Contour: