∫(x² + 4x³ + 2) / (6x) dx

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Understand the Problem

The question involves finding the integral of the expression (x² + 4x³ + 2) divided by 6x with respect to x. This involves applying integration techniques for rational functions.

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 integrand

We start with the expression

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

To simplify, divide each term in the numerator by the denominator:

$$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} $$

So the integral becomes:

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

  1. Integrate each term

Now we integrate each term separately:

  • For the first term,

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

  • For the second term,

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

  • For the third term,

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

Putting it all together, we have:

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

  1. Write the final result

Combine the results into the final answer:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

The final answer is

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

More Information

This result shows how to handle rational functions by simplifying them before integration. Each term integrated separately makes it easier to apply basic integration techniques.

Tips

  • Forgetting to simplify the rational expression before integrating.
  • Mixing up constants when integrating, especially with logarithmic terms.
  • Neglecting the absolute value in the logarithmic term for $x$.

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