Complex integral

Let $C$ denote the unit circle centered at the origin in Complex Plane

What is the value of

$$ \frac{1}{2\pi i}\int_C |1+z+z^2 |dz,$$ where the integral is taken anti-clockwise along $C$?

What I have answered is 0 because it seems like $f(z)$ is analytic at 0 hence by Cauchy's Theorem.

HINT Note that if $z = x+yi$ then $$ \left|1+z+z^2\right| = \left|\left(x^2-y^2+x+1\right) +y(2x+1)i\right| $$ Now parameterize $C$ as $x = \cos t, y = \sin t$ and $t \in [0,2\pi]$, then $$ \int_C |1+z+z^2 |dz = \int_{t=0}^{t=2\pi} \left|\left(x^2-y^2+x+1\right) +y(2x+1)i\right| $$ where $x,y$ are functions of $t$ as in the parameterization. Can you finish this?

Determine if $\sum_{n=1}^\infty \frac{n+4^n}{n+6^n}$ converges

When does a real matrix have a real square root?

Question about Implicit function theorem

Representation of a number as a sum of squares.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

Composition of limits

Why does casting out $9$ and $11$ work to compute remainders?

Given the hierarchy of Borel sets, how to prove that ${\bf \Sigma}_\alpha^0 \subsetneq {\bf \Sigma}_{\alpha+1}^0$ for all ordinal $\alpha<\omega_1$

Give an example where $\int_{\bar A} f$ exists but $\int_A f$ does not for a continuos $f$ on a bounded open subset $A$ of $\mathbb{R}^n $

Derivative of $x^2$

Non integer derivative of $1/p(x)$

Killing form on $\mathfrak{sp}(2n)$