[Math] Billateral laplace transform sin(t) / t u (t)

laplace transform

Hello everyone at my course I have problem solving Laplace transform of

$\frac{\sin(t)}{t}$ $u{(t)}$

I have no idea I tried by definiton but get integral which cant be solved I already took a look at Finding the Laplace Transform of sin(t)/t

But It doesnt help me at all becouse there is used Taylor series expansion
becouse I m still begginer is there any easier way to solve it

Thanks in advante

Best Answer

In THIS ANSWER, I showed that the inverse Laplace Transform of $\arctan(s)-\pi/2$ is $-\frac{\sin(t)}{t}$ by carrying out the integral

$$\mathscr{L}^{-1}(\arctan(s)-\pi/2)=\int_{\sigma -i\infty}^{\sigma+i\infty}e^{st}(\arctan(s)-\pi/2)\,ds$$

Herein, we carry out the forward Laplace Transform of the sinc function. Proceeding, we have

$$\begin{align} F(s)&=\int_0^\infty e^{-st}\frac{\sin(t)}{t}\,dt \end{align}$$

Now, differentiating we have

$$\begin{align} F'(s)&=-\int_0^\infty e^{-st}\sin(t)\,dt\\\\ &=-\frac1{s^2+1} \end{align}$$

whereupon integrating and using $\lim_{s\to \infty }F(s)=0$ yields

$$F(s)=\pi/2-\arctan(s)$$

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