Answer to Question #231364 in Chemical Engineering for pavani

Question #231364

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


1
Expert's answer
2021-09-26T11:59:04-0400

"L\\left\\{\\sinh \\left(\\frac{t}{t}\\right)\\right\\}\\\\\n\\mathrm{Use\\:the\\:linearity\\:property\\:of\\:Laplace\\:Transform:}\\\\\n\\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\\}\\\\\n=-L\\left\\{\\frac{1}{2e}\\right\\}+L\\left\\{\\frac{e}{2}\\right\\}\\\\\nL\\left\\{\\frac{1}{2e}\\right\\}=\\frac{\\frac{1}{2e}}{s}\\\\\n=\\frac{\\frac{1}{2e}}{s}\\\\\n=\\frac{1}{2es}\\\\\nL\\left\\{\\frac{e}{2}\\right\\}=\\frac{\\frac{e}{2}}{s}\\\\\n=\\frac{\\frac{e}{2}}{s}\\\\\n=\\frac{e}{2s}\\\\\n=-\\frac{1}{2es}+\\frac{e}{2s}\\\\\n\\implies \\mathrm{Laplace\\:Transform\\:of\\:}\\sinh \\left(\\frac{t}{t}\\right):\\quad -\\frac{1}{2es}+\\frac{e}{2s}"


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