I don't know whether your x is a function of t which you're trying to find, or whether x is a variable.
Case 1: x is a variable independent of t. Then exp(x) comes out of the integral. If you let C = int_0^1 u(t) dt, then you must solve
C exp(x) = x, or
C = x exp(-x).
Use Lambert's W function to solve.
Case 2: x is a function of t. Then the LHS is a constant (because you've integrated out the only independent variable in the LHS), which implies that the RHS is a constant, and hence x is a constant. Again, pull exp(x) out of the integral, and solve as before.