Evaluating the integral with help of Laplace transforms

I came across an integral that is to be solved by the Laplace transform method. $$\int_0^{\infty} \frac{\sin^2{x}}{x^2}$$ My Approch let $$f(t) = \sin^{2}t$$ and $$\sin^{2}t = \frac{1-\cos2t}{2}$$ also we know from the properties of Laplace transforms $$\frac{\mathcal{L}{f(t)}}{t} = \int_s^\infty F(s) ds$$ so $$\mathcal{L}\left[\frac{1-\cos2t}{2}\right] = \frac{1}{2} \left[ \frac{1}{s} - \frac{s}{s^2+4} \right]$$ now, $$\mathcal{L}\left[\frac{1-\cos2t}{2t}\right] = \frac{1}{2}\int_s^\infty \left(\frac{1}{s} -\frac{s}{s^2+4} \right) ds$$ after integrating $$\mathcal{L} \left[\frac{1-cos2t}{2t}\right]= \frac{1}{2}\left[\ln(s) - \frac{\ln(s^2+4)}{2}\right]\big{|}_s^\infty$$ $$\mathcal{L}\left[\frac{1-\cos2t}{2t}\right] =\frac{1}{2} \left[\ln(\frac{s}{(s^2+4)^{1/2}})\right] \big{|}_s^\infty$$ but at this point the integral becomes not-defined, How can I approach this question as it has to be solved by the method of Laplace transforms Hints are appreciated

First, there's a missing $$2$$ in the denominator of $$\frac{1-\cos 2t}{t}$$. I suppose it's a typo? Next, $$\mathcal{L}\left[\frac{1-\cos2t}{2t}\right] = \frac{1}{2}\int_0^\infty \left(\frac{1}{s} -\frac{s}{s^2+4} \right) ds$$ can't be true because the RHS is possibly a number (the integral actually diverges) while the LHS is a function. What we have instead is $$\mathcal{L}\left[\frac{\sin^2 t}{t}\right]=\mathcal{L}\left[\frac{1-\cos2t}{2t}\right]=\frac 12\int_s^{\infty}\left(\frac 1y-\frac{y}{y^2+4}\right)dy=\frac 14\log\frac{s^2+4}{s^2}$$ Now, if we let $$g(t)=\sin^2 t/t$$, we've found $$\mathcal{L}[g(t)]$$. So $$\int_0^{\infty}\frac{\sin^2 x}{x^2}dx=\int_0^{\infty}\frac{g(x)}{x}dx=\frac 14\int_0^{\infty}\log\frac{p^2+4}{p^2}dp$$ Can you continue from here? Use integration by parts.
• Thanks for your answer, actually i have done an error in the lower limit of the integral integral $\frac{1}{2} \int_s^\infty \left[\frac{1}{s} - \frac{s}{s^2+4} \right] ds$ i have taken the lower limt to $0$ but it should be $s$ and corrected it . But from above answer i have figured out the right method to solve the question. Commented Jan 19, 2021 at 12:55
• @VedantChourey When you write $$\mathcal{L}\left[\frac{f(t)}{t}\right]=\int_s^{\infty}F(s)ds$$ you're using the letter $s$ for $2$ different things: the integration variable and a limit of integration. Use separate letters for these things (for example, $dy$ instead of $ds$). And you've made the same mistake in this comment. Commented Jan 19, 2021 at 12:57