Certain Integrals Arising from Ramanujan's Notebooks

In his third notebook, Ramanujan claims that $${\int }_0^{\mathrm{\infty }}\frac{\cos (nx)}{x^2+1}\log x\mathrm{d}x+\frac{\pi }{2}{\int }_0^{\mathrm{\infty }}\frac{\sin (nx)}{x^2+1}\mathrm{d}x=0.$$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan’s notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by ${\log }^{2}x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples.