In this paper we extend the Dirichlet integral formula of Lobachevsky. Let
f(x) be a continuous function and satisfy in the Ο-periodic assumption
f(x+Ο)=f(x), and f(Οβx)=f(x), 0β€x<β. If the integral
β«0ββx4sin4xβf(x)dx defined in the sense of the improper
Riemann integral, then we show the following equality β«0ββx4sin4xβf(x)dx=β«02Οββf(t)dtβ32ββ«02Οββsin2tf(t)dt
hence if we take f(x)=1, then we have β«0ββx4sin4xβdx=3Οβ Moreover, we give a method for computing
β«0ββx2nsin2nxβf(x)dx for nβNComment: Paper is re-written with new presentation and new results adde