Integral of e^(x+e^(x)) dx

integral(e^(x + e^x) dx) = integral(e^x * e^(e^x) dx). Now use the substitution method:

let u = e^x, then du = e^x dx and the integral becomes

integral(e^u du) = e^u + C = e^(e^x) + C.

∫e^(x + e^(x)) dx = ∫e^(x) * e^(e^(x)) dx = e^(e^(x)) + C