You do it by finding the Laplace transform of the expression. Look at the Frequency integration transform in this link https://en.wikipedia.org/wiki/Laplace_transform , under the Properties and theorems section. So you first find the Laplace of 2(1 - cos(t)), which is 2/s -(2s)/(s^2+1). Then find the ∫ from s to ∞ of 2/σ -(2σ)/(σ^2+1) with respect to dσ, and you will get ln ((s^2 + 1) / s^2) .