I want to find Laplace transform of $\sin(at)$ by definition
$$\displaystyle \mathcal L \left\{{\sin at}\right\} = \int_0^{\to +\infty}e^{-st}\sin at \, \mathrm dt$$
After taking integration by parts twice, I reached the following at last
$$\displaystyle \mathcal L \left\{ {\sin at}\right\}=\displaystyle \left.{-e^{-st} {\frac {s^2}{s^2+a^2} }\left({\frac 1 s \sin at + \frac a {s^2} \cos at}\right)}\right\vert_{t \mathop = 0}^{t \mathop \to +\infty}$$
I can' t do the rest. What is the limit of $\sin at$ or $\cos at$ while $t\rightarrow \infty$
For the rest, Can anybody explain it step by step pls.?
Best Answer
You are forgetting to evaluate $-e^{-st}$ as $t\to\infty$:
$$\mathcal{L}\left\{\sin at\right\}=\frac{s^2}{s^2+a^2} \left[-\color{red}{e^{-st}}\left(\frac{1}{s}\sin at + \frac{a}{s^2}\cos at\right)\right]_{t= 0}^{t=\infty}$$
What happens to the expression in square brackets when $t\to\infty$?