Common Laplace transforms

The following table summarizes common Laplace transforms that can be used to solve different Laplace transform problems:

Function f(t)	Function F(s)=\mathscr{L}\{f(t)\}	Laplace transformation Proof
1	\dfrac{1}{s}	Check proof!
t^{n}	\dfrac{n!}{s^{n+1}},\: n\in\mathbb{Z}_{+}	Check proof!
\sin(bt)	\dfrac{b}{s^{2}+b^{2}}	Check proof!
\cos(bt)	\dfrac{s}{s^{2}+b^{2}}	Check proof!
e^{at}	\dfrac{1}{s-a}	Check proof!
t^{n}e^{at}	\dfrac{n!}{(s-a)^{n+1}},\quad n\in\mathbb{Z}_{+}	Check proof!
e^{at}\sin(bt)	\dfrac{b}{(s-a)^{2}+b^{2}}	Check proof!
e^{at}\cos(bt)	\dfrac{s-a}{(s-a)^{2}+b^{2}}	Check proof!
f(t-a)u(t-a)	e^{-as}F(s)	Check proof!
e^{at}f(t)	F(s-a)	Check proof!
f'(t)	sF(s)-f(0)	Check proof!
f''(t)	s^{2}F(s)-sf'(0)-f(0)	Check proof!
f^{(n)}(t)	s^{n}F(s)-s^{(n-1)}f(0)-\cdots-f^{(n-1)}(0)	Check proof!
t^{n}f(t)	(-1)^{n}\dfrac{d^{n}}{ds^{n}}F(s)
(f\ast g)(t)	(-1)^{n}\dfrac{d^{n}}{ds^{n}}F(s)	Check proof!
\displaystyle\int_{0}^{t}f(\tau)d\tau	\dfrac{1}{s}F(s)	Check proof!
f(at)	\dfrac{1}{a}F\left(\dfrac{s}{a}\right)	Check proof!
f periodic with period T	\dfrac{1}{1-e^{-sT}}\displaystyle\int_{0}^{T}e^{-st}f(t)dt	Check proof!

Download PDF version of common Laplace transforms: