Inverse Laplacetransform, can't find fitting formula

[chubbypaddy]

Homework Statement

Inverse Laplacetransform (3s+7)/(s^2+2s-2)

Homework Equations

(3s+7)/(s^2+2s-2)

The Attempt at a Solution

Split into 3s/(s^2+2s-2) and 7/(s^2+2s-2).

I can't find a fitting transformpair(I use tables/formulas).

 

[GuiltySparks]

Let's complete the square of the polynomial equation to get it into a form that we can use.

s^2 + 2s -2 = (s+1)^2 -3

Now, we have

3s/[(s+1)^2 -3] + 7/[(s+1)^2 -3]
= 3(s+1)/[(s+1)^2 -3] + 4/[(s+1)^2 -3]
invLaplace => 3(e^-t)hcos(rad3*t) + (4/rad3)(e^-t)hsin(rad3*t)

 

[chubbypaddy]

I don't have hyperbolicus functions in my transformtable, is there any other way?

 

[GuiltySparks]

Yes, we can use the definition of sinh and cosh.

Hyperbolic sine: (e^2x - 1)/(2e^x)
Hyperbolic cosine: (e^2x + 1)/(2e^x)

3(e^-t)[(e^2rad3(t) + 1)/(2e^rad3(t))] + (4/rad3)(e^-t)[(e^2rad3(t) - 1)/(2e^rad3(t))]
=3(e^2rad3(t) + 1)/(2e^(rad3(t)+1)) + (4/rad3)(e^2rad3(t) - 1)/(2e^(rad3(t)+1))
=[(3rad3+4)/(2rad3)](e^2rad3(t) - (8/rad3))/(e^(rad3(t)+1))