∫ (x² + 4x³ + 2) / (6x) dx

Question image

Understand the Problem

The question is asking us to solve the integral of a rational function represented by a fraction involving polynomials. We need to simplify and integrate the expression accordingly.

Answer

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
Answer for screen readers

The final answer is:

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

Steps to Solve

  1. Simplify the Expression

First, divide each term in the numerator by the denominator:

$$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} $$

This simplifies to:

$$ \frac{x}{6} + \frac{2x^2}{3} + \frac{1}{3x} $$

  1. Set Up the Integral

Now, set up the integral with the simplified expression:

$$ \int \left( \frac{x}{6} + \frac{2x^2}{3} + \frac{1}{3x} \right) dx $$

  1. Integrate Each Term

Integrate each term separately:

  • For $\frac{x}{6}$:

$$ \int \frac{x}{6} dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$

  • For $\frac{2x^2}{3}$:

$$ \int \frac{2x^2}{3} dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$

  • For $\frac{1}{3x}$:

$$ \int \frac{1}{3x} dx = \frac{1}{3} \ln |x| $$

  1. Combine the Results

Combine all the integrals together:

$$ \int \left( \frac{x}{6} + \frac{2x^2}{3} + \frac{1}{3x} \right) dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

where $C$ is the constant of integration.

The final answer is:

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

More Information

This integral involves basic polynomial integration and the natural logarithm of a linear term. The process showcases how to simplify a rational function before integration, making it easier to manage.

Tips

  • Forgetting the integration constant: Always remember to add the constant of integration ( C ).
  • Improper simplification: Ensure that each term is simplified correctly before integration.
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