# Explain First Shift Theorem & its properties?

Explain First Shift Theorem & its properties?
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Step 1
First Shift Theorem:
If
The proof of the First shift theorem follows from the definition of Laplace transform. It is known that,
$L\left\{f\left(t\right)\right\}={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt$
Step 2
Then,
$L\left\{{e}^{at}f\left(t\right)\right\}={\int }_{0}^{\mathrm{\infty }}{e}^{at}\cdot {e}^{-st}f\left(t\right)dt$
$={\int }_{0}^{\mathrm{\infty }}{e}^{\left(a-s\right)t}f\left(t\right)dt$
$={\int }_{0}^{\mathrm{\infty }}{e}^{-\left(s-a\right)t}f\left(t\right)dt$
$=F\left(s-a\right)$
Thus, if the Laplace transform of function f(t) is known, then we can find the Laplace transform of ${e}^{at}f\left(t\right)$ using First shift theorem.
Step 3
For example,
It is known that, $L\left\{\mathrm{sin}3t\right\}=\frac{3}{\left({s}^{2}+9\right)}$
Then,
$L\left\{{e}^{2t}\cdot \mathrm{sin}3t\right\}=\frac{3}{\left(s-2{\right)}^{2}+9}$ by first shift theorem.