Question Number 219520 by OmoloyeMichael last updated on 27/Apr/25
$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{laplace}}\:\boldsymbol{{transform}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{t}}\right)=\int_{\mathrm{0}} ^{\boldsymbol{{t}}} \frac{\boldsymbol{{sint}}}{\boldsymbol{{t}}}\boldsymbol{{dt}} \\ $$
Answered by SdC355 last updated on 27/Apr/25
$$\int_{\mathrm{0}} ^{\:{T}} \:\:\frac{\mathrm{sin}\left({t}\right)}{{t}}\:\mathrm{d}{t}=\mathrm{Si}\left({T}\right) \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({t}\right)}{{t}}{e}^{−{st}} \:\mathrm{d}{t}=\int_{{s}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({t}\right){e}^{−{xt}} \mathrm{d}{t}\mathrm{d}{x} \\ $$$$\int_{\:{s}} ^{\:\infty} \:\:\frac{\mathrm{d}{x}}{{x}^{\mathrm{2}} +\mathrm{1}}=\frac{\pi}{\mathrm{2}}−\mathrm{tan}^{−\mathrm{1}} \left({s}\right) \\ $$
Commented by OmoloyeMichael last updated on 28/Apr/25
$$\boldsymbol{{thanks}}\:\boldsymbol{{sir}} \\ $$