# Find the integral that includes many level exponents

I tried without success the method by substitution and integration by parts in finding the integral, $$\int e^{-x-e^{-x}} dx.$$

Can somebody provide any tip or propose a solution ?

Thanks.

• This integral does not have a closed form solution. You will need power series or numerical methods. Nov 29 '21 at 15:49
• @EthanBolker That is demonstrably false. In reality, the derivative of $e^{-e^{-x}}$ is the integrand of the exercise. Nov 29 '21 at 16:19
• @Angel Indeed. I jumped hastily to a false conclusion. Nov 29 '21 at 17:04

Notice that you can write the integrand as $$e^{-x}\cdot e^{-e^{-x}}$$. Then use the substitution $$u = -e^{-x}$$. The rest is a straightforward calculation.