Complex Contour Integration using Magnitude and triangle inequality

Question

I have some questions and I have been asked to use the magnitude and triangle inequality. Here is the first question:

Let $C$ be the arc of the circle $|z| = 2$ from $z = 2i$ to $z = 2$, Show that $$\left|\int_C\frac{dz}{z^2-1}\right|\leq\frac{\pi}{3}$$ I'm given this magnitude inequality to be the following theorem:

If on a contour $C$, $|f(z)|\leq M$ and $L$ is the length of $C$, then $$\left|\int_Cf(z)\,dz\right|\leq\int_C|f(z)||dz|\leq ML$$ I would like to have a clear process for this question so I can attempt the few after it on my own. I just don't see what I should be looking for. What to do first.

Answer 1

HINTS:

$$\left|\int_C f(z)\,dz\right|\le \int_C |f(z)|\,|dz|$$

And $|z_1+z_2|\ge ||z_1|-|z_2||$.