# text{Laplace transforms A powerful tool in solving problems inengineering and physics is the Laplace transform.

$\left(F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt$
$\text{where we assume s is a positive real number. For example, to find the Laplace transform of}$
$F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}{e}^{-t}dt={\int }_{0}^{\mathrm{\infty }}{e}^{-\left(s+1\right)t}dt=\frac{1}{s+1}$

$f\left(t\right)=t\to F\left(s\right)=\frac{1}{{s}^{2}}$

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$\text{Step 1}$

$\text{Step 2}$

$\text{Laplace transformation of f(t) is defined as:}$
$L\left(t\right)=F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt$

$L\left(t\right)=F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}tdt$
\(\text{Use integration by part } \int fg=fg-int