Evaluate the integral of (x² + 4x + 2)/(6x) dx.

Question image

Understand the Problem

The question is asking for the evaluation of the integral of a rational function, specifically the expression (x² + 4x + 2)/(6x) with respect to x. This involves performing polynomial long division or integration techniques.

Answer

$$ \int \frac{x^2 + 4x + 2}{6x} \, dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Answer for screen readers

$$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$

Steps to Solve

  1. Simplify the Integral Begin by rewriting the integrand. We can separate the terms in the integrand as follows:

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

This simplifies to:

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

  1. Set up the Integral The integral can then be expressed as the sum of three integrals:

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

  1. Integrate each term Now we integrate each term separately:
  • For the first term, apply the power rule:

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

  • For the second term:

$$ \int \frac{2}{3} dx = \frac{2}{3} x $$

  • For the third term, recall that the integral of $1/x$ is $\ln|x|$:

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

  1. Combine the Results Combine all the parts together to get the final result:

$$ \int \left(\frac{x^2 + 4x + 2}{6x}\right) dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$

where $C$ is the constant of integration.

$$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$

More Information

This integral simplifies using basic rules of integration, illustrating key concepts in calculus such as breaking down rational functions and applying the power rule for integration. The constant $C$ represents the arbitrary constant of integration resulting from indefinite integration.

Tips

  • Forgetting to include the constant of integration: Always remember to add $C$ to your final answer after integrating.
  • Incorrectly simplifying fractions: Make sure to combine terms accurately when splitting into separate integrals.
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