Complex analysis integration method

Question

How do you solve the integral $$\int^\infty_{-\infty}\frac{cos z}{z^2+9}dz$$

If I first find the roots, I get $z=-3i$ and $z=3i$

I also know that $$\int^\infty_{-\infty} f(x) dx=2 \pi i \sum^m_{k=1} res(f,c_k)$$

How do I apply this?

Answer 1

Consider a countour $C = C_1 + C_2$ where $C_1 = [-R, R]$ and $C_2 = \{ R e^{it} : 0 \leqslant t \leqslant \pi \}$ and function $f(z) = \frac{e^{iz}}{z^2+9}$. If you could prove that $$\lim_{R \to \infty} \int \limits_{C_2} f(z) \, \mathrm{d} z = 0,$$ then $$\mathrm{res}( f, 3i ) = \int \limits_C f(z) \, \mathrm{d} z = \lim_{R \to \infty} \left[ \int \limits_{C_1} f(z) \, \mathrm{d} z + \int \limits_{C_2} f(z) \, \mathrm{d} z \right] = \int \limits_{-\infty}^{\infty} f(z) \, \mathrm{d} z = \int \limits_{-\infty}^{\infty} \left( \frac{\cos x}{x^2+9} + i \cdot \frac{\sin x}{x^2+9} \right) \, \mathrm{d} x.$$

Finish by extracting the real part.