# 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?

• @MichaelHardy Please don't add \displaystyle to titles; see here. The title is perfectly readable without it. Oct 17, 2014 at 18:52
• @epimorphic : \displaystyle appeared twice in this title before I edited it. After that it appeared only once. ${}\qquad{}$ Oct 17, 2014 at 19:01
• @MichaelHardy Ah, sorry, I only read the beginning of the part being edited. I guess the point still stands that it should be removed altogether. Oct 17, 2014 at 19:05

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 $\lim_{a \to \infty}\operatorname{Ci}(a)=0$
simply the result is$$\int_0^{\infty} \frac{\cos x}{x} \, dx = \infty$$