Evaluate the integral of (x^2 + 4x + 7) / (6x) dx.

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Understand the Problem

The question is asking for the integral of a rational function, specifically the integral of (x^2 + 4x + 7) / (6x) with respect to x. The approach will involve simplifying the expression and then applying integral rules.

Answer

$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$
Answer for screen readers

The final result of the integral is:

$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$

Steps to Solve

  1. Simplify the integrand

We start by simplifying the expression (\frac{x^2 + 4x + 7}{6x}). We can separate the terms in the numerator:

$$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} $$

This simplifies to:

$$ \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} $$

  1. Set up the integral

Now that we have the simplified expression, we set up the integral:

$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx $$

  1. Integrate each term separately

Now we will integrate each term individually:

  • For (\frac{x}{6}):

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

  • For (\frac{2}{3}):

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

  • For (\frac{7}{6x}):

$$ \int \frac{7}{6x} , dx = \frac{7}{6} \ln |x| $$

  1. Combine the results

Putting it all together, we have:

$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$

where (C) is the constant of integration.

The final result of the integral is:

$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$

More Information

This result gives the antiderivative of the rational function. Integrating rational functions often involves simplifying the expression before performing the integration.

Tips

  • Ignoring the simplification step can lead to complications in integration.
  • Failing to separate the constant factor when integrating each term can result in errors.
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