Question

# use properties of the Laplace transform and the table of Laplace transforms to determine L[f] f(t)=2+2(e^{-t}-1)u_1(t)

Laplace transform
use properties of the Laplace transform and the table of Laplace transforms to determine L[f]
$$f(t)=2+2(e^{-t}-1)u_1(t)$$

2020-12-29
$$\text{Step 1}$$
$$\text{Given, }f(t)=2+2(e^{-t}-1)u_1(t)$$
$$\text{To determine the Laplace Transform } L\left[f(t)\right]$$
$$\text{Step 2}$$
$$L\left[f(t)\right]=L\left[2+2(e^{-t}-1)u_1(t)\right]$$
$$=L\left[2\right]+L\left[2(e^{-t}-1)u_1(t)\right]$$
$$=2L\left[1\right]+2L\left[e^{-t}u_1(t)\right]-2L\left[u_1(t)\right] \text{ using linearity of Laplace Tranform, }$$
$$=2\frac{1}{s}+2e^{-s}L\left[e^{-t}\right]-2\frac{e^{-s}}{s}$$
$$=\frac{2}{s}+\frac{2e^{-s}}{s+1}-\frac{2e^{-s}}{s}$$