integral from 0 to infinity of exp(exp(-x))*ln(x) dx

The integral in question is:

\[
\int_{0}^{\infty} e^{e^{-x}} \ln(x) \, dx
\]

This integral is non-trivial and can be evaluated using special functions, specifically the **Gamme Function** and its derivatives.

To solve this, we can start by considering the series expansion of the exponential function and then interchange the order of summation and integration. The solution involves the digamma function \(\psi(x)\), which is the derivative of the logarithm of the Gamma function \(\Gamma(x)\).

### Detailed Solution:

1. **Series Expansion of Exponential**:
   
   \[
   e^{e^{-x}} = \sum_{n=0}^{\infty} \frac{(e^{-x})^n}{n!} = \sum_{n=0}^{\infty} \frac{e^{-nx}}{n!}
   \]

2. **Substitute into the Integral**:

   \[
   \int_{0}^{\infty} \ln(x) \, e^{e^{-x}} \, dx = \int_{0}^{\infty} \ln(x) \left( \sum_{n=0}^{\infty} \frac{e^{-nx}}{n!} \right) dx
   \]
   
   Interchanging the summation and integration:

   \[
   \sum_{n=0}^{\infty} \frac{1}{n!} \int_{0}^{\infty} \ln(x) \, e^{-nx} \, dx
   \]

3. **Use Known Integral Result**:

   The integral \( \int_{0}^{\infty} \ln(x) \, e^{-nx} \, dx \) can be evaluated using the relationship involving the digamma function:

   \[
   \int_{0}^{\infty} \ln(x) \, e^{-nx} \, dx = -\frac{1}{n} \psi(1)
   \]

   where \(\psi(1) = -\gamma\), and \(\gamma\) is the Euler-Mascheroni constant.

4. **Final Expression**:

   Substituting this result into the series:

   \[
   -\sum_{n=1}^{\infty} \frac{\psi(1)}{n^2 n!}
   \]
   
   Simplifying using \(\psi(1) = -\gamma\):

   \[
   \gamma \sum_{n=1}^{\infty} \frac{1}{n^2 n!}
   \]

   Recognizing that the sum converges to a known value:

   \[
   \int_{0}^{\infty} e^{e^{-x}} \ln(x) \, dx = \gamma \times \zeta(2) = \gamma \times \frac{\pi^2}{6}
   \]

### Final Answer:

\[
\int_{0}^{\infty} e^{e^{-x}} \ln(x) \, dx = \frac{\gamma \pi^2}{6}
\]

Would you like more details on any of the steps or have any questions?

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Here are 5 related questions:

1. What is the digamma function and how is it related to the Gamma function?
2. Can the series expansion method be applied to other types of integrals?
3. What is the significance of the Euler-Mascheroni constant \(\gamma\)?
4. How can we derive the sum \(\sum_{n=1}^{\infty} \frac{1}{n^2 n!}\)?
5. What other types of special functions are useful in evaluating integrals?

**Tip:** Understanding series expansions and special functions is crucial for solving complex integrals involving non-elementary functions.

Learn how to evaluate the integral from 0 to infinity of exp(exp(-x))*ln(x) dx using series expansion and special functions. This detailed solution involves the gamma function, digamma function, and the Euler-Mascheroni constant. Suitable for advanced undergraduate students and mathematicians interested in special functions and integral calculus.

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