Asymptotic expansion of cosine integral Can anybody help with this problem to find the full asymptotic expansion of 
$\int_1^\infty \frac{\cos(xt)\, d t}{t}$ from Bender & Orzsag). Does it work by Taylor  expansion?

In this case, integration by parts is the method of choice. You can show easily that (for $n>0$)
$$\int_1^\infty\!dt\, \frac{\cos(x t)}{t^n} = -\frac{\sin(x)}{x} + \frac{n}{x} \int_1^\infty\!dt\, \frac{\sin(x t)}{t^{n+1}} .$$
Similarly
$$\int_1^\infty\!dt\, \frac{\sin(x t)}{t^n} = \frac{\cos(x)}{x} - \frac{n}{x}\int_1^\infty\!dt\, \frac{\cos(x t)}{t^{n+1}}. $$
Applying these results recursively, we obtain
$$\int_1^\infty\!dt\, \frac{\cos(x t)}{t^1} = -\frac{\sin(x)}{x} + \frac{1}{x} \int_1^\infty\!dt\, \frac{\sin(x t)}{t^{2}}
=-\frac{\sin(x)}{x}+ \frac{\cos(x)}{x^2} - \frac{2}{x^2}\int_1^\infty\!dt\, \frac{\cos(x t)}{t^{3}} = \dots \\
= \cos(x)\left(\frac{1}{x^2} -\frac{3!}{x^4} + \frac{5!}{x^6} - \dots\right) - \sin(x) \left(\frac{1}{x} -\frac{2!}{x^3} + \frac{4!}{x^5} - \dots\right)$$
