Question

Let \[ f(t)= \begin{cases} \sin t, & 0\le t\le \pi,\\ 0, & t>\pi. \end{cases} \] The Laplace transform of \(f(t)\) is

• A. \(\dfrac{1}{s^2+1}\)
• B. \(\dfrac{e^{-\pi s}}{s^2+1}\;(s>0)\)
• C. \(\dfrac{e^{-\pi s}+1}{s^2+1}\;(s>0)\)
• D. \(\dfrac{e^{-\pi s}+e^{-\pi}}{s^2+1}\;(s>0)\)

Hint:

Whenever a function is zero after a finite interval, \[ \mathcal{L}\{f(t)\} = \int_0^{a}e^{-st}f(t)\,dt, \] where \(a\) is the point beyond which the function becomes zero.

Answer 1

The Correct Option is C