Integrate $\int_0^{\infty } \frac{\cos x}{x} dx$

Although I have known that $\displaystyle\int_0^\infty {{\sin x} \over x} \, dx = {\pi \over 2}$, I have no idea how to work out $\displaystyle\int_0^{ + \infty } {{\cos x} \over x} \, dx$. How can I?

Unfortunately, the integral $$ \int_0^\infty\frac{\cos x}{x}\,dx, $$ diverges.

To see this observe that $$ \int_0^{\pi/4}\frac{\cos x}{x}\,dx\ge \frac{\sqrt{2}}{2}\int_0^{\pi/4}\frac{1}{x}=\infty. $$

$$\int \frac{\cos x}{x} \, dx = \operatorname{Ci}(x) + C,$$ where $\operatorname{Ci}$ is the Cosine Integral.

Your integral does not converge on the interval $[0,\infty)$, since $\operatorname{Ci}(0)=-\infty$ and $\lim_{a \to \infty}\operatorname{Ci}(a)=0$