Properties of Laplace transform
1. Linearity Property
$$ A f_1(t) + B f_2(t) \longleftrightarrow A F_1(s) + B F_2(s) $$
2. Frequency Shifting Property
$$ e^{s_0t} f(t)) \longleftrightarrow F(s – s_0) $$
3. nth Derivative Property
$$ \frac {d^n f(t)}{dt^n} \longleftrightarrow s^n F(s) − \sum_{i=0}^n s^{n − i} f^{i − 1} (0^−) $$
4. Integration
$$ \int\limits_0^t f(\lambda) \ d\lambda \longleftrightarrow \frac 1s F(s) $$
5. Multiplication by Time
$$ T f(t) \longleftrightarrow −\frac {d F(s)}{ds} $$
6. Complex Shift Property
$$ f(t) e^{−at} \longleftrightarrow F(s + a) $$
7. Time Reversal Property
$$ f (-t) \longleftrightarrow F(-s) $$
8. Time Scaling Property
$$ f \left(\frac ta\right) \longleftrightarrow a F(as) $$