# Inverse Laplace Transform

1. Mar 10, 2012

### TyErd

1. The problem statement, all variables and given/known data
Find the inverse laplace transform of $\frac{3s + 7}{s^{2} - 2s + 10}$

2. Relevant equations
completing the square.
$e^{at}sin(bt) = \frac{b}{(s-a)^{2} + b^{2}}$
$e^{at}cos(bt) = \frac{s-a}{(s-a)^{2} + b^{2}}$

3. The attempt at a solution
F(s)= $\frac{3s + 7}{s^{2} - 2s + 10}$
F(s) = $\frac{3s + 7}{(s-1)^{2} +9}$
F(s) = $\frac{3s}{(s-1)^{2} +9} + \frac{7}{(s-1)^{2} +9}$

after this i don't know how to manipulate the first fraction to fit the cosine equation. I know the 3 can be taken up front and a=1 and b=3 im pretty sure when comparing with the cosine equation but there the problem of making s into s-1.

2. Mar 10, 2012

### LCKurtz

Don't you have the shifting theorems? Like$$\mathcal L e^{at}f(t) = \mathcal L(f(t))|_{s \to s-a}$$