Editing Dirichlet integral (section)

=== Double integration ===

Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the [[Order of integration (calculus)|order of integration]], namely,
<math display="block">
\left( I_1 = \int_0^\infty \int _0^\infty e^{-st} \sin t \,dt \,ds \right) = \left( I_2 = \int_0^\infty \int _0^\infty e^{-st} \sin t \,ds \,dt \right),</math>
<math display="block">\left( I_1 = \int_0^\infty \frac{1}{s^2 + 1} \,ds = \frac{\pi}{2} \right) = \left( I_2 = \int_0^\infty \frac{\sin t}{t} \,dt \right), \text{ provided } s > 0.
</math>
The change of order is justified by the fact that for all <math>s > 0</math>, the integral is absolutely convergent.