Find the Laplace Transform

Question

Could anyone enlighten me on how to find the Laplace Transform of

$$\frac{1-\cos (t)}{t}$$

Answer 1

Hint:

Set $$f(t):=\frac{1-\cos t}{t}.$$ We have $$\mathcal{L}\{1-\cos t \}(s)=\mathcal{L} \{tf(t) \}(s)=-F'(s), $$ where $F(s)$ is the Laplace transform of $f(t)$.

Answer 2

$$ f(t)=\frac{g(t)}{t}=\frac{1-\cos t}{t} $$ thus the Laplace transform is $$ F(s)=\int_s^\infty G(z)\,\mathrm dz=\int_s^\infty \left[\frac{1}{z}-\frac{z}{z^2+1}\right]\,\mathrm dz=\left[\log{z}-\log\left(\sqrt{z^2+1}\right)\right]_s^\infty=\log\left(\tfrac{\sqrt{s^2+1}}{s}\right) $$