Evaluate $\int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$

Question

Problem.¹⁾ Evaluate $\displaystyle \int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$, given that $\mathcal{C} : |z| = \pi/2$

Answer 1

Use Cauchy's integral formula.

$$\int_{|z|=a} \frac{f(z)}{z-a}dz = 2\pi i f(a) $$

For your question $f(z) = sin(z)$

So answer to your question is $2\pi i sin(\frac{\pi}{2}) = 2\pi i$