Nonlinear equations with exponent

Problem 6

The function y=f(a)y = f(a) is the solution of the equation

2y=ay. 2^{-y} = ay.

The function F(x)F(x) is the integral

F(x)=0xf(t)dt. F(x) = \int_0^{x} f(t)dt.

Find the value of the integral

010F(x)dx! \int_0^{10} F(x)dx!