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{\bf Exam Question

Topic: Laplace}

The function $f$ is defined in terms of the function $g$ by
$$f(x)=\int_0^x g(t)dt.$$ Write down the definition of the Laplace
transform of $f(x).$ this gives a repeated integral. Reverse the
order of integration and evaluate the inner integral.

Deduce that $L(g)=pL(f),$ where $L$ denotes the Laplace transform.
\vspace{0.5in}

{\bf Solution}

\begin{eqnarray*}
L(f(x)&=&\int_0^{\infty}\mathrm{e}^{-px}\int_0^x g(t)\, dt =
\int_0^{\infty}\, dt\int_t^{\infty}\mathrm{e}^{-px}g(t)\, dt\\
&=&\frac{1}{p}\int_0^{\infty}\mathrm{e}^{-pt}g(t)\, dt =
\frac{1}{p}L(g).\\ \mathrm{So}\ \ L(g)&=& pL(f).
\end{eqnarray*}


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