Question

Laplace Transformation: t-u_(pi/2)(t)(t-(pi/2))

Answer 1

To make things clearer I'll write out the eqn here:\[t-u _{\pi/2}(t)(t-(\pi/2))\]

Answer 2

I'm just double checking, but would it be:\[(1-e^{-\pi/2})/s^2\]

Answer 3

\[f(t)=t-u_{\pi/2}(t)(t-\pi/2)\]\[F(s)=\int\limits_0^{+\infty}f(t)e^{-st}dt=\int\limits_0^{+\infty}te^{-st}dt-\int\limits_0^{+\infty}u_{\pi/2}(t)(t-\pi/2)e^{-st}dt=\]\[=\frac{1}{s^2}-\int\limits_{\pi/2}^{+\infty}(t-\pi/2)e^{-st}dt=\frac{1}{s^2}-\int\limits_{0}^{+\infty}ue^{-s(u+\pi/2)}du=\]\[=\frac{1}{s^2}-e^{-s\pi/2}\int\limits_{0}^{+\infty}ue^{-su}du=\frac{1}{s^2}-e^{-s\pi/2}\frac{1}{s^2}=\]\[=\frac{1}{s^2}(1-e^{-s\pi/2})\]