table of Laplace transform pairs
Nobody re-derives the same integrals every time they use the Laplace transform. Instead there is a short table — a two-column dictionary that lists common time functions f(t) on the left and their transforms F(s) on the right. You look up the piece you need, the way you once memorized that the derivative of sin is cos.
The core entries are worth knowing by heart: 1 maps to 1/s; t maps to 1/s^2 and more generally t^n to n!/s^{n+1}; e^{a t} maps to 1/(s - a); sin(omega t) maps to omega/(s^2 + omega^2) and cos(omega t) to s/(s^2 + omega^2). The hyperbolic versions, sinh and cosh, give omega/(s^2 - omega^2) and s/(s^2 - omega^2). The table is read in both directions: left-to-right is the forward transform, and right-to-left is the inverse transform, which is how you finish a Laplace-method calculation. The whole table is generated by a handful of base pairs plus the structural rules.
What makes a small table so powerful is that you almost never use it alone. Combined with linearity and the shifting theorems, a handful of entries covers an enormous range: the first shifting theorem turns the e^{a t} entry into the transforms of e^{a t} cos(omega t) and e^{a t} sin(omega t) for free, and after a partial-fraction decomposition any rational F(s) is just a sum of these standard pieces. The table is the destination of every inverse-transform problem.
From the table: L{t^2} = 2/s^3 (using n! with n = 2) and L{cos(3 t)} = s/(s^2 + 9). Reading right-to-left, the inverse of 5/(s^2 + 9) is (5/3) sin(3 t).
Each entry runs both ways; the inverse direction is where partial fractions deliver their answer.
A table is only as good as the convergence assumptions baked into it: every entry quietly assumes t >= 0 and an implicit factor of the Heaviside step, so transforming a function defined for negative t with a plain table entry is a mistake.