A lower bound for the cosine integral

Question

I am reading Devroye's paper 2001Simulating Perpetuities. On P103, he mentioned a lower bound for cosine integral, i.e., $$\int_0^t\frac{1-\cos s}{s}ds \geq max(0,\gamma+\log t),$$ where $\gamma$ is 0.5 less than Euler's constant.

I checked the reference he cited, but did not see how I can get to the equation. Wondering someone could help me on this.

Answer 1

$$\int\frac{1-\cos (s)}{s}\,ds =\log (s)-\text{Ci}(s)$$ $$\int_0^t\frac{1-\cos (s)}{s}\,ds =\gamma+\log (t)-\text{Ci}(t)$$ and the largest value of $\text{Ci}(t)$ is about $0.47200$