sibuzwaW

2021-01-13

The function
$\left\{\begin{array}{ll}t& 0\le t<1\\ {e}^{t}& t\ge 1\end{array}$
has the following Laplace transform,
$L\left(f\left(t\right)\right)={\int }_{0}^{1}t{e}^{-st}dt+{\int }_{1}^{\mathrm{\infty }}{e}^{-\left(s+1\right)t}dt$
True or False

Nathanael Webber

Expert

Step 1
Definition used -
Laplace transform of a function f(t) is given by -
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}dt$
Step 2
Given -
$\left\{\begin{array}{ll}t& 0\le t<1\\ {e}^{t}& t\ge 1\end{array}$
$L\left(f\left(t\right)\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}dt$
$={\int }_{0}^{1}t{e}^{-st}dt+{\int }_{1}^{\mathrm{\infty }}{e}^{t}{e}^{-st}dt$
$={\int }_{0}^{1}t{e}^{-st}dt+{\int }_{1}^{\mathrm{\infty }}{e}^{\left(1-s\right)t}dt$
Step 3
Answer - So given Laplace transform is False for function.

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