Math Problem Statement

\int Ln(x) dx

Solution

To find the integral of ln(x)\ln(x), we can use integration by parts. The formula for integration by parts is:

udv=uvvdu\int u \, dv = uv - \int v \, du

For the integral ln(x)dx\int \ln(x) \, dx, we can choose:

  • u=ln(x)u = \ln(x)
  • dv=dxdv = dx

Now, we need to find dudu and vv:

  • du=1xdxdu = \frac{1}{x} \, dx
  • v=xv = x

Using the integration by parts formula:

ln(x)dx=xln(x)x1xdx\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx

This simplifies to:

ln(x)dx=xln(x)1dx\int \ln(x) \, dx = x \ln(x) - \int 1 \, dx

ln(x)dx=xln(x)x+C\int \ln(x) \, dx = x \ln(x) - x + C

where CC is the constant of integration.

Would you like more details or have any questions?

Here are five related questions you might want to ask:

  1. How do you integrate exln(x)e^x \ln(x)?
  2. What is the integral of ln(x)2\ln(x)^2?
  3. How do you use integration by parts in other contexts?
  4. Can you explain the process of substitution in integration?
  5. What are some common applications of the natural logarithm function?

Tip: Integration by parts is particularly useful for integrals involving the natural logarithm, products of polynomials and exponentials, or products of trigonometric functions and exponentials.

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Natural Logarithm

Formulas

Integration by parts

Theorems

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Suitable Grade Level

Undergraduate