Find the Laplace transform of the following funtions

Question

f(t)=t(sinh t + cosh t)²

Answer 1

Probably the best thing you can do to start this problem is rewrite, (sinh t + cosh t)². Using the definition of

sinh(t) = (e^t-e^-t)/2 and cosh(t) = (e^t+e^-t)/2,

(sinh t + cosh t)²=e^2x.

Given L[tⁿe^at] = n!/(s-a)ⁿ⁺¹, then

L[t(sinh t + cosh t)²] = 1/(s-2)².