Laplace Transform (idea)

Here's a table of some basic Laplace transforms.

            function            Laplace
      
            1                   1/s
      
            eat                 1/ (s-a)
      
            tn                  n! / s (n + 1) 
      
            tp, p >-1           Γ(p + 1) / s (p + 1) 
      
            sin at              a / (s 2  + a 2 )
      
            cos at              s / (s 2  + a 2 )
      
            sinh at             a / (s 2  - a 2 )
      
            cosh at             s / (s 2  - a 2 )
      
            e at sin bt          b / [ (s-a) 2  + b 2 ]
      
            e at cos bt          (s - a) / [ (s-a) 2  + b 2 ]
      
            tneat                n! / (s - a) (n + 1) 
      
            uc(t)                e -cs / s
      
            uc(t)f(t-c)          e-csF(s)
      
            ect f(t)             F(s - c)
      
            f(ct)                1/c F(s/c)
      
            ∫0tf(t-τ)g(τ)dτ      F(s)*G(s)
      
            δ(t - c)             e-cs 
      
            f(n)(t)              snF(s)-sn-1f(0)-...-f(n - 1)(0)
      
            (-t)nf(t)            F(n)(s)

For notation clarity, I used u_c(t) in this table to denote the unit step function, or Heaviside function (this is often H(t-c) in some differential equations books).