Lobachevsky integral formula

In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions.
One of those is the improper integral of the sinc function over the positive real line,
Lobachevsky's Dirichlet integral formula.
Let formula_2 be a continuous function satisfying the formula_3-periodic assumption formula_4, and formula_5, for formula_6. If the integral formula_7 is taken to be an improper Riemann integral, we have Lobachevsky's Dirichlet integral formula
Moreover, we have the following identity as an extension of the Lobachevsky Dirichlet integral formula
As an application, take formula_10. Then