two improper integrals
Let us consider first the improper integral
The derivative may be formed by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign):
Here, the last form has been gotten by the substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) . But since by the parent entry (http://planetmath.org/SineIntegralInInfinity) we have
and since , we can write
Thus we have evaluated the integral :
(1) |
The formula (1) gives
We use here the consequence formula
of the double angle formula , obtaining
where the substitution has produced the last form. Accordingly, we can write as result the formula
(2) |