Why Does My Laplace Transform Calculation Differ from the Textbook's Result?

[Eastonc2]

Homework Statement

f(t)=e^(t+7)

Homework Equations

£{f(t)}=∫e^(-st)f(t)dt

The Attempt at a Solution

so i insert my f(t) into the formula, came up with ∫e^(-st+t+7)dt
using u substitution, u=t(-s+1)+7, du=(-s+1)dt so it follows that 1/(-s+1)∫e^(u)du=e^(u)/(-s+1)
so I plug u back in, and should be able to find my answer from there, only I come up with an answer very different from the one in the book, which is e^(7)/(s-1)
Can anyone help me out?

So i figured it out, I set u=-1(t(-s+1)+7)=t(s-1)-7, and put e^-u inside the integral. turns out just making myself look at it a little longer worked out

 

[LCKurtz]

Homework Statement

f(t)=e^(t+7)

Homework Equations

£{f(t)}=∫e^(-st)f(t)dt

The Attempt at a Solution

so i insert my f(t) into the formula, came up with ∫e^(-st+t+7)dt
using u substitution, u=t(-s+1)+7, du=(-s+1)dt so it follows that 1/(-s+1)∫e^(u)du=e^(u)/(-s+1)
so I plug u back in, and should be able to find my answer from there, only I come up with an answer very different from the one in the book, which is e^(7)/(s-1)
Can anyone help me out?

You didn't show your work with the limits, which is where your error is. You need to either put the new u limits in your u answer or the t limits in your t answer.

 

[Eastonc2]

ok, so I figured out that last one, now I'm having difficulty with £{t^(2)e^(-2t)}. putting it into the laplace definition I come up with ∫t^(2)e^(-t(s+2))dt. I've tried integration by parts, and come up with:
u=t^2, du=2t dt, dv=e^(-t(s+2))dt

and that's where i get stuck, i can't seem to figure out this integration. I've plugged it into wolfram, but that turns out with v=te^(-t(s+2)), which, when plugged back into the integration by parts, leaves me with a more complicated equation, involving the negative of my original integral.

 

[6.28318531]

You can always use integration by parts again,

 

[Eastonc2]

changed a few things around, still using integration by parts, actually integrated by parts i think three times total, if I'm remembering last night correctly, to finally end up with the right answer. thanks