This is a piece wise function. I'm not sure how to do piece wise functions in latex.
$$ f(t) =\begin{cases}\sin t &\text{if } 0 \le t < \pi,\\
0&\text{if } t \ge \pi.\end{cases} $$
So we want to take the Laplace transform of that equation.
So I get $L\{\sin t\} + L\{0\}$
Using the Laplace identities I get $L\{\sin t\} = \frac{1}{s^{2} + 1}$.
And $L\{0\} = 0$.
So for my answer I get $\frac{1}{s^{2} + 1} + 0$.
But the answer in the back of the book is $$\frac{1 + e^{-\pi s}}{s^{2} + 1}.$$
Where does the $$\frac{e^{- \pi s}}{s^{2} + 1}$$ come from?
Best Answer
Hint: You need to evaluate the integral
$$\int_{0}^{\infty}e^{-st}f(t)dt=\int_{0}^{\pi}e^{-st}\sin t dt$$
Do not use your tables in this case.