∫ (x² + 4x + 2) / (6x) dx

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Understand the Problem

The question is asking for the evaluation of the integral of a rational function, specifically the integral of (x² + 4x + 2) / (6x) with respect to x. This involves simplifying the expression and then applying the appropriate integration techniques.

Answer

The integral evaluates to: $$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers

The final result of the integral is:

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

Steps to Solve

  1. Simplify the expression

To simplify the expression $\frac{x^2 + 4x + 2}{6x}$, we can separate the terms in the numerator:

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

This simplifies to:

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

  1. Set up the integral

Now we set up the integral of the simplified expression:

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

  1. Integrate term by term

Next, we integrate each term separately:

  • For $\int \frac{x}{6} , dx$:

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

  • For $\int \frac{2}{3} , dx$:

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

  • For $\int \frac{1}{3x} , dx$:

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

  1. Combine the results

Combining all the results from the integration:

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

where $C$ is the constant of integration.

The final result of the integral is:

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

More Information

This integral represents the area under the curve defined by the rational function over a specified interval. Integrating rational functions often involves breaking them down into simpler components, which makes the integration process more manageable.

Tips

  • Forgetting to simplify the rational expression before integrating.
  • Misapplying the rules of integration, particularly for logarithmic integrals.
  • Not including the constant of integration $C$.

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