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

The integral of the natural logarithm function ln(x) is a fundamental result in calculus. It's commonly used in various areas of mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the standard integral formula:

Where:

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

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

3. Importance of ln(x) Integral

Details: This integral appears frequently in probability theory (entropy calculations), thermodynamics, and when solving differential equations involving logarithmic terms.

4. Using the Calculator

Tips: Enter a positive x value (x > 0) and optionally specify the integration constant C. The calculator will compute x*ln(x) - x + C.

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.

Q2: What is the integration constant C?
A: The constant represents the family of all antiderivatives. Its value is determined by initial conditions in practical applications.

Q3: Can this be used for definite integrals?
A: Yes, subtract the value at the lower limit from the value at the upper limit (the constant cancels out).

Q4: What if I need to integrate ln(ax) instead?
A: Use the substitution rule: ∫ln(ax)dx = xln(ax) - x + C.

Q5: How is this integral derived?
A: Using integration by parts with u = ln(x) and dv = dx, giving du = (1/x)dx and v = x.