\int_a^b f(x,\alpha)\hbox{d}x:=\int_a^c f(x,\alpha)\hbox{d}x+\int_c^b f(x,\alpha)\hbox{d}x=\lim_{\eta\to a_+}\int_\eta^c f(x,\alpha)\hbox{d}x+\lim_{\xi\to b_-}\int_c^\xi f(x,\alpha)\hbox{d}x