1. What is the Integral of ln(x)?

The integral of the natural logarithm function ln(x) is a fundamental result in calculus. It appears frequently in mathematical analysis and applications involving logarithmic growth or decay.

2. How Does the Calculator Work?

The calculator uses the standard formula:

Where:

• \( \ln(x) \) — Natural logarithm of x
• \( x \) — Variable of integration
• \( C \) — Constant of integration

Explanation: This result comes from integration by parts, where we let u = ln(x) and dv = dx.

3. Importance of the Natural Logarithm Integral

Details: This integral is essential in probability theory (entropy calculations), physics (thermodynamics), and engineering (signal processing). It's also the basis for more complex logarithmic integrals.

4. Using the Calculator

Tips: Enter a positive x value (x > 0) and optionally specify the integration constant C. The calculator will compute the value of the antiderivative at your specified point.

5. Frequently Asked Questions (FAQ)

Q1: Why is x limited to positive values?
A: The natural logarithm ln(x) is only defined for x > 0, so its integral is likewise only defined for positive x.

Q2: What does the integration constant C represent?
A: The constant accounts for the family of all antiderivatives. In applications, it's often determined by initial conditions.

Q3: Can this be used for definite integrals?
A: Yes, subtract the values at the endpoints: ∫[a,b] ln(x) dx = (b ln(b) - b) - (a ln(a) - a).

Q4: How is this related to the exponential function?
A: The integral of 1/x is ln(x), showing the deep connection between logarithms and exponential functions.

Q5: Are there applications in real-world problems?
A: Yes, this integral appears in information theory (entropy calculations), economics (logarithmic utility), and thermodynamics.