These are commonly used Laplace Transform identities for solving problems efficiently without recomputing integrals.
\[ \delta(t) \iff 1 \quad ; \quad 1 \iff \frac{1}{s} \quad (s > 0) \]
\[ t^n \iff \frac{n!}{s^{n+1}} \quad (n > -1) \]
\[ t^{1/2} \iff \frac{\sqrt{\pi}}{4s^{3/2}} \quad ; \quad t^{-1/2} \iff \frac{\sqrt{\pi}}{s^{1/2}} \]
\[ e^{at} \iff \frac{1}{s - a} \quad ; \quad te^{at} \iff \frac{1}{(s - a)^2} \]
\[ (1 - at)e^{-at} \iff \frac{s}{(s + a)^2} \quad ; \quad t^2e^{-at} \iff \frac{2}{(s + a)^3} \]
\[ \sin(at) \iff \frac{a}{s^2 + a^2} \quad ; \quad \cos(at) \iff \frac{s}{s^2 + a^2} \]
\[ \sinh(at) \iff \frac{a}{s^2 - a^2} \quad ; \quad \cosh(at) \iff \frac{s}{s^2 - a^2} \]
\[ e^{-bt} \sin(at) \iff \frac{a}{(s + b)^2 + a^2} \]
\[ e^{-bt} \cos(at) \iff \frac{s + b}{(s + b)^2 + a^2} \]
\[ e^{-at}f(t) \iff F(s + a) \]