∫ (x^2 + 4x + 2) / (6x) dx

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Understand the Problem

The question is asking for the integral of the function (x^2 + 4x + 2) / (6x) with respect to x. We will solve this by simplifying the expression and then performing the integration.

Answer

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

The integral is

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

Steps to Solve

  1. Simplify the integrand

We start by simplifying the expression ( \frac{x^2 + 4x + 2}{6x} ). We can split the fraction into separate terms:

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

This simplifies to:

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

  1. Rewrite the integral

Now we can rewrite the integral with the simplified expression:

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

  1. Integrate term by term

Next, we will integrate each term individually:

  • The integral of ( \frac{x}{6} ) is:

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

  • The integral of ( \frac{2}{3} ) is:

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

  • The integral of ( \frac{1}{3x} ) is:

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

  1. Combine the results

Combining these results, we get:

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

where ( C ) is the constant of integration.

The integral is

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

More Information

This integral represents the area under the curve of the function ( \frac{x^2 + 4x + 2}{6x} ) and is useful in various applications, such as finding the total distance traveled by an object over time.

Tips

  • Forgetting to split the fraction before integrating.
  • Misapplying the integral rules, especially for constants and logarithmic functions.
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