$L\left[\sin3t\right],\ s\rightarrow s+2$  

 $L\left[\sin3t\right],\ s\rightarrow s-2$  

Linearity Property

First Shifting Property

Convolution Theorem

Second Shifting Property

Linearity Property

First Shifting Property

Convolution Theorem

Second Shifting Property

 $a=2,\ f\left(t\right)=e^{2t}$  

 $a=4,\ f\left(t\right)=\cos\ t$  

 $a=2,\ f\left(t\right)=\sin\ t$  

 $\frac{2}{\left(s-3\right)^4}-\left(\frac{3}{\left(\left(s-3\right)^3\right)}\right)+\frac{5}{\left(s-3\right)^2}$  

 $\frac{6}{\left(s-3\right)^4}-\left(\frac{6}{\left(s-3\right)^3}\right)+\frac{5}{\left(s-3\right)^2}$  

 $\frac{2}{\left(s-3\right)^3}-\left(\frac{3}{\left(s-3\right)^{\left(2\right)}}\right)+\frac{5}{s-3}$  

Linearity Property

Derivative of Laplace Transform

First Shifting Property

Second Shifting Property

 $n=1,\ f\left(t\right)=\sin\ 4t,\ F\left(s\right)=\frac{4}{s^2+16}$  

 $n=4,\ f\left(t\right)=t,\ F\left(s\right)=\frac{4}{s^2+16}$  

 $n=1,\ f\left(t\right)=\cos\ 6t,\ F\left(s\right)=\frac{s}{s^2+14}$  

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

$ℒ\left(𝑓′(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

 $G\left(s\right)=F\left(s-3\right)$  

 $G\left(s\right)=\left(-1\right)^3\frac{\text{d}^3F\left(s\right)}{\text{d}s^3}$  

 $G\left(s\right)=F\left(s+3\right)$  

 $G\left(s\right)=\left(-1\right)^2\frac{\text{d}^2F\left(s\right)}{\text{d}s^2}$  

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

 $ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$ 

$\frac{a}{s^2-a^2}$

$\frac{a}{s^2+a^2}$

$\frac{4}{s^2-16}$

$\frac{4}{s^2+16}$

 $G\left(s\right)=\int_s^{\infty}F\left(u\right)du$  

 $G\left(s\right)=\left(-1\right)\frac{\text{d}F\left(s\right)}{\text{d}s}$  

 $G\left(s\right)=\int_{\infty}^sF\left(u\right)du$  

 $G\left(s\right)=^{\left(-1\right)^{-1}}\frac{\text{d}F\left(s\right)}{\text{d}s}$