считаем внутренний интеграл
[blue]∫ _(0)^(1)e^(2x)\cdot e^(2y)dy[/blue]dx=e^(2x) ∫ e^(2y)*(1/2)d(2y)=(1/2)e^(2x)(e^(2y))|^(1)_(0)=
=(1/2)e^(2x)(e^(2*1)-e^(0))=(1/2)e^(2x)(e^2-1)
тогда
∫ _(0)^(1)[blue](∫ _(0)^(1)e^(2x)\cdot e^(2y)dy[/blue])dx= ∫ _(0)^(1)[blue](1/2)e^(2x)(e^2-1)[/blue]dx=
=((e^2-1)/2)*∫ _(0)^(1)[blue]e^(2x)[/blue]dx=[замена t=2x; x=t/2; dx=(1/2)dt]=
=((e^2-1)/2)*∫ _(0)^(1)[blue]e^(2x)[blue]dx=((e^2-1)/2)* (1/2)∫ _(0)^(2)[blue]e^(t)[blue](1/2)dt=
=((e^2-1)/4)*(e^(t))|^(2)_(0)=((e^2-1)/4)*(e^2-1)=(e^2-1)^2/4)