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)

Question
Laplace transform
asked 2020-12-28
use properties of the Laplace transform and the table of Laplace transforms to determine L[f]
\(f(t)=2+2(e^{-t}-1)u_1(t)\)

Answers (1)

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