Finding the Laplace Transform of a Polynomial Function

[kidi3]

Homework Statement

Find the Laplace transform of the following function
t2 - 2t

The Attempt at a Solution

\[\begin{gathered}<br /> f\left( t \right) = {t^2} - 2t \hfill \\<br /> l\left( {f\left( t \right)} \right) = F\left( s \right) = \int_0^\infty {{e^{ - st}} \cdot \left( {{t^2} - 2t} \right)ds} \hfill \\<br /> \Leftrightarrow \hfill \\<br /> = \int_0^\infty {{e^{ - st}}{t^2} - 2t{e^{ - st}}ds} \hfill \\<br /> = \int_0^\infty {{e^{ - st}}{t^2}} ds - \int_0^\infty {2t{e^{ - st}}} ds \hfill \\<br /> = \int_0^\infty {{e^{ - st}}{t^2}} ds - \int_0^\infty {2t{e^{ - st}}} ds \hfill \\<br /> {\text{Integration by parts}} \hfill \\<br /> \int_0^\infty {{e^{ - st}}{t^2}} ds \hfill \\<br /> \int_{}^{} {uv&#039; = uv - \int_{}^{} {u&#039;v} } \hfill \\<br /> v&#039;\left( t \right) = {e^{ - st}} \Rightarrow v = \frac{{ - 1}}{s}{e^{ - st}} \hfill \\<br /> u\left( t \right) = {t^2} \Rightarrow u&#039;\left( t \right) = 2t \hfill \\<br /> \int_{}^{} {uv&#039; = uv - \int_{}^{} {u&#039;v} } \hfill \\<br /> \int_0^\infty {{e^{ - st}}{t^2}} ds = \frac{{ - 1}}{s}{e^{ - st}} \cdot {t^2} - \int_{}^{} {2t \cdot } \frac{{ - 1}}{s}{e^{ - st}}ds = \frac{{ - 1}}{s}{e^{ - st}} \cdot {t^2} - \frac{{ - 2}}{s}\int_{}^{} {t \cdot } {e^{ - st}}ds \hfill \\ <br /> \end{gathered} \]am i doing it correctly?

 

[Karnage1993]

You're supposed to integrate with respect to $t$, not $s$! Switch the $ds$ with $dt$ and start over.

 

[kidi3]

Ahh.. It makes sense now.

 

[kidi3]

Could you tell me what i am doing wrong with this one..

Think i got it..