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

To solve the integral \(\int_0^\infty e^{-x} \ln(x) \, dx\), we can use a technique involving the **Gamma function** and the **digamma function**. Here's the step-by-step solution:

### Step 1: Recognize the Integral as a Special Form
The given integral can be related to the following known form:

\[
\int_0^\infty x^{a-1} e^{-x} \ln(x) \, dx = \Gamma'(a)
\]

where \(\Gamma'(a)\) is the derivative of the Gamma function \(\Gamma(a)\) with respect to \(a\). This derivative is also related to the **digamma function** \(\psi(a)\), where:

\[
\Gamma'(a) = \Gamma(a) \psi(a)
\]

### Step 2: Specific Case \(a = 1\)
For the given integral, \(a = 1\), so:

\[
\int_0^\infty e^{-x} \ln(x) \, dx = \Gamma(1) \psi(1)
\]

### Step 3: Evaluate Gamma and Digamma Functions
- \(\Gamma(1) = 1\)
- \(\psi(1)\) is the digamma function at 1, which is known to be \(-\gamma\), where \(\gamma\) is the **Euler-Mascheroni constant**.

Thus:

\[
\int_0^\infty e^{-x} \ln(x) \, dx = 1 \times (-\gamma) = -\gamma
\]

### Final Answer
\[
\int_0^\infty e^{-x} \ln(x) \, dx = -\gamma
\]

where \(\gamma \approx 0.5772\).

Would you like more details, or do you have any questions?

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Here are 5 related questions:
1. How is the Gamma function defined and what are its properties?
2. What is the digamma function and how does it relate to the Gamma function?
3. Can this integral be solved using integration by parts? If so, how?
4. What is the significance of the Euler-Mascheroni constant in mathematics?
5. How do the Gamma and digamma functions appear in probability theory?

**Tip:** Understanding special functions like Gamma and digamma can greatly help in solving complex integrals in mathematical analysis.

Learn how to solve the integral \(\int_0^\infty e^{-x} \ln(x) \, dx\) using the Gamma function and the digamma function. Detailed step-by-step solution with insights into special functions. Suitable for advanced mathematics enthusiasts and students.

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