inverse oflaplace 5/((s^2+5^2)^2)

{ "query": { "display": "inverse laplace $$\\frac{5}{\\left(s^{2}+5^{2}\\right)^{2}}$$", "symbolab_question": "LAPLACE#inverselaplace \\frac{5}{(s^{2}+5^{2})^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Laplace", "subTopic": "Inverse", "default": "\\frac{1}{50}(\\sin(5t)-5t\\cos(5t))" }, "steps": { "type": "interim", "title": "Laplace Inverse Transform of $$\\frac{5}{\\left(s^{2}+5^{2}\\right)^{2}}:{\\quad}\\frac{1}{50}\\left(\\sin\\left(5t\\right)-5t\\cos\\left(5t\\right)\\right)$$", "input": "L^{-1}\\left\\{\\frac{5}{\\left(s^{2}+5^{2}\\right)^{2}}\\right\\}", "steps": [ { "type": "step", "result": "=L^{-1}\\left\\{\\frac{1}{50}\\cdot\\:\\frac{2\\cdot\\:5^{3}}{\\left(s^{2}+5^{2}\\right)^{2}}\\right\\}" }, { "type": "step", "primary": "Use the constant multiplication property of Inverse Laplace Transform:<br/>For function $$f\\left(t\\right)$$ and constant $$a:{\\quad}L^{-1}\\{a{\\cdot}f\\left(t\\right)\\}=a{\\cdot}L^{-1}\\{f\\left(t\\right)\\}$$", "result": "=\\frac{1}{50}L^{-1}\\left\\{\\frac{2\\cdot\\:5^{3}}{\\left(s^{2}+5^{2}\\right)^{2}}\\right\\}" }, { "type": "step", "primary": "Use Inverse Laplace Transform table: $$L^{-1}\\{\\frac{2a^{3}}{\\left(s^{2}+a^{2}\\right)^{2}}\\}=\\sin\\left(at\\right)-at\\cos\\left(at\\right)$$", "secondary": [ "$$L^{-1}\\{\\frac{2\\cdot\\:5^{3}}{\\left(s^{2}+5^{2}\\right)^{2}}\\}=\\sin\\left(5t\\right)-5t\\cos\\left(5t\\right)$$" ], "result": "=\\frac{1}{50}\\left(\\sin\\left(5t\\right)-5t\\cos\\left(5t\\right)\\right)" } ], "meta": { "solvingClass": "Laplace" } } }