Question

\[\begin{align*}
f(x)&=\frac{\sin(ax)}x\\
\\F(p)&=\mathcal L\left\{\frac{\sin(ax)}x\right\}\\
\\&=\int_p^\infty\mathcal L\left\{\sin(ax)\right\} \cdot\text dp'\\
\\&=\int_p^\infty\frac{a}{p'^2+a^2} \cdot\text dp'\\
\\&=\left.\arctan\left(\frac {p'}a\right) \right|_p^\infty\\
\\&=\frac \pi2-\arctan\left(\frac{p}{a}\right)\\
\end{align*}\]