Question

Use transform definitions, and the evaluation
of a suitable double integral, to calculate the
Laplace transform of the following integral
\[\mathcal L\left\{\int\limits_0^tx(u)\cdot\text du\right\}\]

Answer 1

\[\begin{align*}\mathcal L\left\{\int\limits_0^tx(u)\cdot\text du\right\}
&=\int\limits_0^\infty\left(\int\limits_0^tx(u)\cdot\text du\right)e^{-pt}\cdot\text dt\\
&=\\
&=\\
&=\end{align*}\]

Answer 2

I'm guessing that this involve integration by parts...

Answer 3

do i have to reverse the order of integration ?

Answer 4

Not quite sure though.

Answer 5

@Zarkon

Answer 6

\[\begin{align*}\mathcal L\left\{\int\limits_0^tx(u)\cdot\text du\right\}
&=\int\limits_0^\infty\left(\int\limits_0^tx(u)\cdot\text du\right)e^{-pt}\cdot\text dt\\
&=\int\limits_0^\infty\int\limits_0^ux(u)e^{-pt}\cdot\text dt\cdot\text du\\
?&=\int\limits_0^\infty x(u)\int\limits_0^ue^{-pt}\cdot\text dt\cdot\text du\\
\end{align*}\]

Answer 7

@mahmit2012

Answer 8

Somehow u become a t