For determining of the value of the improper integral $$I(a) \;:=\; \int_0^\infty\frac{\cos{ax^2}-\cos{ax}}{x}\,dx \qquad (a > 0),$$ related to the cosine integral, we think it as a function of the parametre $a$ which we denote by $t$ . Then we can take the Laplace transform (see the integration with respect to a parametre in the table of Laplace transforms): $$\mathcal{L}\{I(t)\} \;=\; \mathcal{L}\{\int_0^\infty(\cos{tx^2}-\cos{tx})\frac{dx}{x}\} \;=\; \int_0^\infty\left(\frac{s}{s^2\!+\!x^4}-\frac{s}{s^2\!+\!x^2}\right)\frac{dx}{x}$$ Splitting the fractional expressions to partial fractions and integrating give