#### JJBladester

Gold Member

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**1. Homework Statement**

Find [tex]f(t)[/tex].

**2. Homework Equations**

[tex]L^{-1}\left\{\frac{s}{s^{2}+4s+5}\right\}[/tex]

**3. The Attempt at a Solution**

I tried completing the square to get to the solution and I ended up with:

[tex]L^{-1}\left\{\frac{s}{s^{2}+4s+5}\right\}[/tex] =

[tex]L^{-1}\left\{\frac{s}{(s+2)^{2}+1}\right\}[/tex]

Then, I used the inverse of a transform for cosine and the first translation theorum:

[tex]coskt = L^{-1}\left\{\frac{s}{(s^{2}+k^{2})}\right\}[/tex]

[tex]L\left\{e^{at}f(t)\right\} = F(s-a)}[/tex]

with [tex]a[/tex] being -2 and [tex]k[/tex] being 1 to get an answer of:

[tex]e^{-2t}cos(t)[/tex]

However, I was wrong. The book had the following answer:

[tex]e^{-2t}cos(t)-2e^{-2t}sin(t)[/tex]

**My question is where does the [tex]-2e^{-2t}sin(t)[/tex] come from?**