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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.

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

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    $\begingroup$ Thanks. It solved the issue. $\endgroup$
    – user996159
    Nov 29 '21 at 16:48

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