application of sine integral at infinity
For finding the value of the improper integral
(1) |
we first use the partial fraction representation (http://planetmath.org/PartialFractionsOfExpressions)
Thus we may write
But by the entry sine integral at infinity, the first integral equals . When we check
we see that there is the linear differential equation
(2) |
i.e.
satisfied by the sought function . We have the initial conditions
Therefore the general solution
of (2) requires that , , and consequently the sought integral has the value
(3) |