Question

1)Find the Laplace transform of sinh t/t?

Answer 1

$L\left\{\sinh \left(\frac{t}{t}\right)\right\}\\ \mathrm{Use\:the\:linearity\:property\:of\:Laplace\:Transform:}\\ \mathrm{For\:functions\:}f\left(t\right),\:g\left(t\right)\mathrm{\:and\:constants\:}a,\:b:\quad L\left\{a\cdot f\left(t\right)+b\cdot g\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}+b\cdot L\left\{g\left(t\right)\right\}\\ =-L\left\{\frac{1}{2e}\right\}+L\left\{\frac{e}{2}\right\}\\ L\left\{\frac{1}{2e}\right\}=\frac{\frac{1}{2e}}{s}\\ =\frac{\frac{1}{2e}}{s}\\ =\frac{1}{2es}\\ L\left\{\frac{e}{2}\right\}=\frac{\frac{e}{2}}{s}\\ =\frac{\frac{e}{2}}{s}\\ =\frac{e}{2s}\\ =-\frac{1}{2es}+\frac{e}{2s}\\ \implies \mathrm{Laplace\:Transform\:of\:}\sinh \left(\frac{t}{t}\right):\quad -\frac{1}{2es}+\frac{e}{2s}$