If $$\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} } $$, then $$\alpha$$ + $$\beta$$ is equal to ____________.

The number of real roots of the equation e^4x $$-$$ e^3x $$-$$ 4e^2x $$-$$ e^x + 1 = 0 is equal to ______________.

Let y = y(x) be the solution of the differential equation dy = e^{$$\alpha$$x + y} dx; $$\alpha$$ $$\in$$ N. If y(log_e2) = log_e2 and y(0) = log_e$$\left( {{1 \over 2}} \right)$$, then the value of $$\alpha$$ is equal to _____________.

Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number (10)¹⁰ . (11)¹¹ . (13)¹³ is equal to __________.