# Show that $|\int_{C} \frac{1}{z^{3}+1}\, dz|\leq \frac{\pi}{3}(\frac{R}{R^{3}-1})$

Could someone help me through this problem?

Let C be an arc of the circle $|z|=R$, with $R>1$ of angle $\frac{\pi}{3}$.

Show that $\left|\displaystyle\int_{C} \frac{1}{z^{3}+1}\, dz\right|\leq \dfrac{\pi}{3}\left(\dfrac{R}{R^{3}-1}\right)$

and deduce $\lim\limits_{R \to{+}\infty}{\displaystyle\int_{C} \frac{1}{z^{3}+1}\, dz}$

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Hint: For any contour $\Gamma$ one has that
$$\left|\int_\Gamma f(z)\;dz\right|\leqslant \|f\|_\infty|\Gamma|$$
Where $|\Gamma|$ is the arc-length of $\gamma$.
As a supplement to this hint, for any complex $a$ and $b$ one has $|a - b| \geq ||a| - |b||$. So in particular, if $|z| = R > 1$, putting $a = z^3$ and $b = -1$ one deduces that $|z^3 + 1| \geq R^3 - 1$. –  leslie townes Apr 26 '12 at 3:13