 # Can the Inverse Finding the Laplace transform of (2s+1)/(s(s+1)(s+2)) without using partial fractions? Jaylen Dudley 2022-09-16 Answered
Can the Inverse Finding the Laplace transform of $\frac{2s+1}{s\left(s+1\right)\left(s+2\right)}$ without using partial fractions?
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It is not right away the convolution of two functions but you can split into two fractions and use convolution on each one and add the results .

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$F\left(s\right)=\frac{2s+1}{s\left(s+1\right)\left(s+2\right)}$
With $s\to s-\frac{1}{2}$

then from $F\left(s-c\right)={e}^{cx}f\left(x\right)$ we have
$f\left(x\right)={e}^{-\frac{1}{2}x}\left(-{e}^{-x}-\frac{1}{2}{e}^{-2x}+\frac{3}{2}\right)=-{e}^{-\frac{3}{2}x}-\frac{1}{2}{e}^{-\frac{5}{2}x}+\frac{3}{2}{e}^{-\frac{1}{2}x}$

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